Probability is a measure that is associated with how certain we are of outcomes of a particular experiment or activity. An **experiment **is a planned operation carried out under controlled conditions. If the result is not predetermined, then the experiment is said to be a **chance** experiment. Flipping one fair coin twice is an example of an experiment.

A result of an experiment is called an **outcome**. The **sample space** of an experiment is the set of all possible outcomes. Three ways to represent a sample space are: to list the possible outcomes, to create a tree diagram, or to create a Venn diagram. The uppercase letter *S* is used to denote the sample space. For example, if you flip one fair coin, (S = { ext{H, T}}) where ( ext{H} =) heads and ( ext{T} =) tails are the outcomes.

An **event **is any combination of outcomes. Upper case letters like ( ext{A}) and ( ext{B}) represent events. For example, if the experiment is to flip one fair coin, event ( ext{A}) might be getting at most one head. The probability of an event ( ext{A}) is written (P( ext{A})).

Definition: probability

The *probability *of any outcome is the long-term relative frequency of that outcome. Probabilities are between zero and one, inclusive (that is, zero and one and all numbers between these values).

- (P( ext{A}) = 0) means the event ( ext{A}) can never happen.
- (P( ext{A}) = 1) means the event ( ext{A}) always happens.
- (P( ext{A}) = 0.5) means the event ( ext{A}) is equally likely to occur or not to occur. For example, if you flip one fair coin repeatedly (from 20 to 2,000 to 20,000 times) the relative frequency of heads approaches 0.5 (the probability of heads).

**Equally likely** means that each outcome of an experiment occurs with equal probability. For example, if you toss a **fair**, six-sided die, each face (1, 2, 3, 4, 5, or 6) is as likely to occur as any other face. If you toss a fair coin, a Head (( ext{H})) and a Tail (( ext{T})) are equally likely to occur. If you randomly guess the answer to a true/false question on an exam, you are equally likely to select a correct answer or an incorrect answer.

**To calculate the probability of an event A when all outcomes in the sample space are equally likely**, count the number of outcomes for event ( ext{A}) and divide by the total number of outcomes in the sample space. For example, if you toss a fair dime and a fair nickel, the sample space is ({ ext{HH, TH, HT,TT}}) where ( ext{T} =) tails and ( ext{H} =) heads. The sample space has four outcomes. ( ext{A} =) getting one head. There are two outcomes that meet this condition ( ext{{HT, TH}}), so (P( ext{A}) = frac{2}{4} = 0.5).

Suppose you roll one fair six-sided die, with the numbers {1, 2, 3, 4, 5, 6} on its faces. Let event ( ext{E} =) rolling a number that is at least five. There are two outcomes {5, 6}. (P( ext{E}) = frac{2}{6}). If you were to roll the die only a few times, you would not be surprised if your observed results did not match the probability. If you were to roll the die a very large number of times, you would expect that, overall, (frac{2}{6}) of the rolls would result in an outcome of "at least five". You would not expect exactly (frac{2}{6}). The long-term relative frequency of obtaining this result would approach the theoretical probability of (frac{2}{6}) as the number of repetitions grows larger and larger.

Definition: law of large numbers

This important characteristic of probability experiments is known as the law of large numbers which states that as the number of repetitions of an experiment is increased, the relative frequency obtained in the experiment tends to become closer and closer to the theoretical probability. Even though the outcomes do not happen according to any set pattern or order, overall, the long-term observed relative frequency will approach the theoretical probability. (The word **empirical** is often used instead of the word observed.)

It is important to realize that in many situations, the outcomes are not equally likely. A coin or die may be **unfair**, or **biased**. Two math professors in Europe had their statistics students test the Belgian one Euro coin and discovered that in 250 trials, a head was obtained 56% of the time and a tail was obtained 44% of the time. The data seem to show that the coin is not a fair coin; more repetitions would be helpful to draw a more accurate conclusion about such bias. Some dice may be biased. Look at the dice in a game you have at home; the spots on each face are usually small holes carved out and then painted to make the spots visible. Your dice may or may not be biased; it is possible that the outcomes may be affected by the slight weight differences due to the different numbers of holes in the faces. Gambling casinos make a lot of money depending on outcomes from rolling dice, so casino dice are made differently to eliminate bias. Casino dice have flat faces; the holes are completely filled with paint having the same density as the material that the dice are made out of so that each face is equally likely to occur. Later we will learn techniques to use to work with probabilities for events that are not equally likely.

The "OR" Event

An outcome is in the event ( ext{A OR B}) if the outcome is in ( ext{A}) or is in ( ext{B}) or is in both ( ext{A}) and ( ext{B}). For example, let ( ext{A} = {1, 2, 3, 4, 5}) and ( ext{B} = {4, 5, 6, 7, 8}). ( ext{A OR B} = {1, 2, 3, 4, 5, 6, 7, 8}). Notice that 4 and 5 are NOT listed twice.

The "AND" Event

An outcome is in the event ( ext{A AND B}) if the outcome is in both ( ext{A}) and ( ext{B}) at the same time. For example, let ( ext{A}) and ( ext{B}) be {1, 2, 3, 4, 5} and {4, 5, 6, 7, 8}, respectively. Then ( ext{A AND B} = {4, 5}).

The **complement **of event ( ext{A}) is denoted ( ext{A'}) (read "A prime"). ( ext{A'}) consists of all outcomes that are **NOT **in ( ext{A}). Notice that (P( ext{A}) + P( ext{A′}) = 1). For example, let ( ext{S} = {1, 2, 3, 4, 5, 6}) and let ( ext{A} = {1, 2, 3, 4}). Then, ( ext{A′} = {5, 6}). (P(A) = frac{2}{6}), (P( ext{A′}) = frac{2}{6}), and (P( ext{A}) + P( ext{A′}) = frac{4}{6} + frac{2}{6} = 1)

The conditional probability of ( ext{A}) given ( ext{B}) is written (P( ext{A|B})). (P( ext{A|B})) is the probability that event ( ext{A}) will occur given that the event ( ext{B}) has already occurred. **A conditional reduces the sample space**. We calculate the probability of ( ext{A}) from the reduced sample space ( ext{B}). The formula to calculate (P( ext{A|B})) is (P( ext{A|B}) = frac{ ext{P(A AND B)}}{ ext{P(B)}}) where (P( ext{B})) is greater than zero.

For example, suppose we toss one fair, six-sided die. The sample space ( ext{S} = {1, 2, 3, 4, 5, 6}). Let ( ext{A} =) face is 2 or 3 and ( ext{B} =) face is even (2, 4, 6). To calculate (P( ext{A|B})), we count the number of outcomes 2 or 3 in the sample space ( ext{B} = {2, 4, 6}). Then we divide that by the number of outcomes ( ext{B}) (rather than ( ext{S})).

We get the same result by using the formula. Remember that ( ext{S}) has six outcomes.

[P( ext{A|B}) = dfrac{ ext{ P(A AND B) } } {P( ext{B})} = dfrac{dfrac{ ext{(the number of outcomes that are 2 or 3 and even in S)}}{6}}{dfrac{ ext{(the number of outcomes that are even in S)}}{6}} = dfrac{dfrac{1}{6}}{dfrac{3}{6}} = dfrac{1}{3}]

## Understanding Terminology and Symbols

It is important to read each problem carefully to think about and understand what the events are. Understanding the wording is the first very important step in solving probability problems. Reread the problem several times if necessary. Clearly identify the event of interest. Determine whether there is a condition stated in the wording that would indicate that the probability is conditional; carefully identify the condition, if any.

Example (PageIndex{1})

The sample space (S) is the whole numbers starting at one and less than 20.

- (S =) _____________________________
Let event (A =) the even numbers and event (B =) numbers greater than 13.

- (A =) _____________________, (B =) _____________________
- (P( ext{A}) =) _____________, (P( ext{B}) =) ________________
- ( ext{A AND B} =) ____________________, ( ext{A OR B} =) ________________
- (P( ext{A AND B}) =) _________, (P( ext{A OR B}) =) _____________
- ( ext{A′} =) _____________, (P( ext{A′}) =) _____________
- (P( ext{A}) + P( ext{A′}) =) ____________
- (P( ext{A|B}) =) ___________, (P( ext{B|A}) =) _____________; are the probabilities equal?

**Answer**

- ( ext{S} = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19})
- ( ext{A} = {2, 4, 6, 8, 10, 12, 14, 16, 18}, ext{B} = {14, 15, 16, 17, 18, 19})
- (P( ext{A}) = frac{9}{19}), (P( ext{B}) = frac{6}{19})
- ( ext{A AND B} = {14,16,18}), ( ext{A OR B} = {2, 4, 6, 8, 10, 12, 14, 15, 16, 17, 18, 19})
- (P( ext{A AND B}) = frac{3}{19}), (P( ext{A OR B}) = frac{12}{19})
- ( ext{A′} = 1, 3, 5, 7, 9, 11, 13, 15, 17, 19); (P( ext{A′}) = frac{10}{19})
- (P( ext{A}) + P( ext{A′}) = 1left((frac{9}{19} + frac{10}{19} = 1 ight))
- (P( ext{A|B}) = frac{ ext{P(A AND B)}}{ ext{P(B)}} = frac{3}{6}, P( ext{B|A}) = frac{ ext{P(A AND B)}}{ ext{P(A)}} = frac{3}{9}), No

Exercise (PageIndex{1})

The sample space *S* is the ordered pairs of two whole numbers, the first from one to three and the second from one to four (Example: (1, 4)).

- (S =) _____________________________

Let event (A =) the sum is even and event (B =) the first number is prime. - (A =) _____________________, (B =) _____________________
- (P( ext{A}) =) _____________, (P( ext{B}) =) ________________
- ( ext{A AND B} =) ____________________, ( ext{A OR B} =) ________________
- (P( ext{A AND B}) =) _________, (P( ext{A OR B}) =) _____________
- ( ext{B′} =) _____________, (P( ext{B′)} =) _____________
- (P( ext{A}) + P( ext{A′}) =) ____________
- (P( ext{A|B}) =) ___________, (P( ext{B|A}) =) _____________; are the probabilities equal?

**Answer**

- ( ext{S} = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)})
- ( ext{A} = {(1,1), (1,3), (2,2), (2,4), (3,1), (3,3)})

( ext{B} = {(2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)}) - (P( ext{A}) = frac{1}{2}), (P( ext{B}) = frac{2}{3})
- ( ext{A AND B} = {(2,2), (2,4), (3,1), (3,3)})

( ext{A OR B} = {(1,1), (1,3), (2,1), (2,2), (2,3), (2,4), (3,1), (3,2), (3,3), (3,4)}) - (P( ext{A AND B}) = frac{1}{3}, P( ext{A OR B}) = frac{5}{6})
- ( ext{B′} = {(1,1), (1,2), (1,3), (1,4)}, P( ext{B′}) = frac{1}{3})
- (P( ext{B}) + P( ext{B′}) = 1)
- (P( ext{A|B}) = frac{P( ext{A AND B})}{P( ext{B})} = frac{1}{2}, P( ext{B|A}) = frac{P( ext{A AND B})}{P( ext{B})} = frac{2}{3}), No.

Example (PageIndex{2A})

A fair, six-sided die is rolled. Describe the sample space *S*, identify each of the following events with a subset of *S* and compute its probability (an outcome is the number of dots that show up).

- Event ( ext{T} =) the outcome is two.
- Event ( ext{A} =) the outcome is an even number.
- Event ( ext{B} =) the outcome is less than four.
- The complement of ( ext{A}).
- ( ext{A GIVEN B})
- ( ext{B GIVEN A})
- ( ext{A AND B})
- ( ext{A OR B})
- ( ext{A OR B′})
- Event ( ext{N} =) the outcome is a prime number.
- Event ( ext{I} =) the outcome is seven.

**Solution**

- ( ext{T} = {2}), (P( ext{T}) = frac{1}{6})
- (A = {2, 4, 6}), (P( ext{A}) = frac{1}{2})
- ( ext{B} = {1, 2, 3}), (P( ext{B}) = frac{1}{2})
- ( ext{A′} = {1, 3, 5}, P( ext{A′}) = frac{1}{2})
- ( ext{A|B} = {2}), (P( ext{A|B}) = frac{1}{3})
- ( ext{B|A} = {2}), (P( ext{B|A}) = frac{1}{3})
- ( ext{A AND B} = {2}, P( ext{A AND B}) = frac{1}{6})
- ( ext{A OR B} = {1, 2, 3, 4, 6}), (P( ext{A OR B}) = frac{5}{6})
- ( ext{A OR B′} = {2, 4, 5, 6}), (P( ext{A OR B′}) = frac{2}{3})
- ( ext{N} = {2, 3, 5}), (P( ext{N}) = frac{1}{2})
- A six-sided die does not have seven dots. (P(7) = 0).

Example (PageIndex{2B})

Table describes the distribution of a random sample (S) of 100 individuals, organized by gender and whether they are right- or left-handed.

Right-handed | Left-handed | |
---|---|---|

Males | 43 | 9 |

Females | 44 | 4 |

Let’s denote the events (M =) the subject is male, (F =) the subject is female, (R =) the subject is right-handed, (L =) the subject is left-handed. Compute the following probabilities:

- (P( ext{M}))
- (P( ext{F}))
- (P( ext{R}))
- (P( ext{L}))
- (P( ext{M AND R}))
- (P( ext{F AND L}))
- (P( ext{M OR F}))
- (P( ext{M OR R}))
- (P( ext{F OR L}))
- (P( ext{M'}))
- (P( ext{R|M}))
- (P( ext{F|L}))
- (P( ext{L|F}))

**Answer**

- (P( ext{M}) = 0.52)
- (P( ext{F}) = 0.48)
- (P( ext{R}) = 0.87)
- (P( ext{L}) = 0.13)
- (P( ext{M AND R}) = 0.43)
- (P( ext{F AND L}) = 0.04)
- (P( ext{M OR F}) = 1)
- (P( ext{M OR R}) = 0.96)
- (P( ext{F OR L}) = 0.57)
- (P( ext{M'}) = 0.48)
- (P( ext{R|M}) = 0.8269) (rounded to four decimal places)
- (P( ext{F|L}) = 0.3077) (rounded to four decimal places)
- (P( ext{L|F}) = 0.0833)

## Chapter Review

In this module we learned the basic terminology of probability. Events are subsets of the sample space, and they are assigned a probability that is a number between zero and one, inclusive.

## Formula Review

( ext{A}) and ( ext{B}) are events

(P( ext{S}) = 1) where ( ext{S}) is the sample space

(0 leq P( ext{A}) leq 1)

(P( ext{A|B}) = frac{ ext{P(A AND B)}}{ ext{P(B)}})

## Glossary

- Conditional Probability
- the likelihood that an event will occur given that another event has already occurred

- Equally Likely
- Each outcome of an experiment has the same probability.

- Event
- a subset of the set of all outcomes of an experiment; the set of all outcomes of an experiment is called a
**sample space**and is usually denoted by (S). An event is an arbitrary subset in (S). It can contain one outcome, two outcomes, no outcomes (empty subset), the entire sample space, and the like. Standard notations for events are capital letters such as (A, B, C), and so on.

- Experiment
- a planned activity carried out under controlled conditions

- Outcome
- a particular result of an experiment

- Probability
- a number between zero and one, inclusive, that gives the likelihood that a specific event will occur; the foundation of statistics is given by the following 3 axioms (by A.N. Kolmogorov, 1930’s): Let (S) denote the sample space and (A) and (B) are two events in
*S*. Then: - (0 leq P( ext{A}) leq 1)
- If ( ext{A}) and ( ext{B}) are any two mutually exclusive events, then ( ext{P}( ext{A OR B}) = P( ext{A}) + P( ext{B})).
- (P( ext{S}) = 1)

- Sample Space
- the set of all possible outcomes of an experiment

- The AND Event
- An outcome is in the event ( ext{A AND B}) if the outcome is in both ( ext{A AND B}) at the same time.

- The Complement Event
- The complement of event ( ext{A}) consists of all outcomes that are NOT in ( ext{A}).

- The Conditional Probability of
*A*GIVEN*B* - (P( ext{A|B})) is the probability that event ( ext{A}) will occur given that the event ( ext{B}) has already occurred.

- The Or Event
- An outcome is in the event ( ext{A OR B}) if the outcome is in ( ext{A}) or is in ( ext{B}) or is in both ( ext{A}) and ( ext{B}).

Exercise 3.2.2

In a particular college class, there are male and female students. Some students have long hair and some students have short hair. Write the **symbols** for the probabilities of the events for parts a through j. (Note that you cannot find numerical answers here. You were not given enough information to find any probability values yet; concentrate on understanding the symbols.)

- Let ( ext{F}) be the event that a student is female.
- Let ( ext{M}) be the event that a student is male.
- Let ( ext{S}) be the event that a student has short hair.
- Let ( ext{L}) be the event that a student has long hair.

- The probability that a student does not have long hair.
- The probability that a student is male or has short hair.
- The probability that a student is a female and has long hair.
- The probability that a student is male, given that the student has long hair.
- The probability that a student has long hair, given that the student is male.
- Of all the female students, the probability that a student has short hair.
- Of all students with long hair, the probability that a student is female.
- The probability that a student is female or has long hair.
- The probability that a randomly selected student is a male student with short hair.
- The probability that a student is female.

**Answer**

- (P( ext{L′)} = P( ext{S}))
- (P( ext{M OR S}))
- (P( ext{F AND L}))
- (P( ext{M|L}))
- (P( ext{L|M}))
- (P( ext{S|F}))
- (P( ext{F|L}))
- (P( ext{F OR L}))
- (P( ext{M AND S}))
- (P( ext{F}))

*Use the following information to answer the next four exercises.* A box is filled with several party favors. It contains 12 hats, 15 noisemakers, ten finger traps, and five bags of confetti.

Let (H =) the event of getting a hat.

Let (N =) the event of getting a noisemaker.

Let (F =) the event of getting a finger trap.

Let (C =) the event of getting a bag of confetti.

Exercise 3.2.3

Find (P( ext{H})).

Exercise 3.2.4

Find (P( ext{N})).

**Answer**

(P( ext{N}) = frac{15}{42} = frac{5}{14} = 0.36)

Exercise 3.2.5

Find (P( ext{F})).

Exercise 3.2.6

Find (P( ext{C})).

**Answer**

(P( ext{C}) = frac{5}{42} = 0.12)

*Use the following information to answer the next six exercises.* A jar of 150 jelly beans contains 22 red jelly beans, 38 yellow, 20 green, 28 purple, 26 blue, and the rest are orange.

Let (B =) the event of getting a blue jelly bean

Let (G =) the event of getting a green jelly bean.

Let (O =) the event of getting an orange jelly bean.

Let (P =) the event of getting a purple jelly bean.

Let (R =) the event of getting a red jelly bean.

Let (Y =) the event of getting a yellow jelly bean.

Exercise 3.2.7

Find (P( ext{B})).

Exercise 3.2.8

Find (P( ext{G})).

**Answer**

(P( ext{G}) = frac{20}{150} = frac{2}{15} = 0.13)

Exercise 3.2.9

Find (P( ext{P})).

Exercise 3.2.10

Find (P( ext{R})).

**Answer**

(P( ext{R}) = frac{22}{150} = frac{11}{75} = 0.15)

Exercise 3.2.11

Find (P( ext{Y})).

Exercise 3.2.12

Find (P( ext{O})).

**Answer**

(P(text{O}) = frac{150-22-38-20-28-26}{150} = frac{16}{150} = frac{8}{75} = 0.11)

*Use the following information to answer the next six exercises.* There are 23 countries in North America, 12 countries in South America, 47 countries in Europe, 44 countries in Asia, 54 countries in Africa, and 14 in Oceania (Pacific Ocean region).

Let ( ext{A} =) the event that a country is in Asia.

Let ( ext{E} =) the event that a country is in Europe.

Let ( ext{F} =) the event that a country is in Africa.

Let ( ext{N} =) the event that a country is in North America.

Let ( ext{O} =) the event that a country is in Oceania.

Let ( ext{S} =) the event that a country is in South America.

Exercise 3.2.13

Find (P( ext{A})).

Exercise 3.2.14

Find (P( ext{E})).

**Answer**

(P( ext{E}) = frac{47}{194} = 0.24)

Exercise 3.2.15

Find (P( ext{F})).

Exercise 3.2.16

Find (P( ext{N})).

**Answer**

(P( ext{N}) = frac{23}{194} = 0.12)

Exercise 3.2.17

Find (P( ext{O})).

Exercise 3.2.18

Find (P( ext{S})).

**Answer**

(P( ext{S}) = frac{12}{194} = frac{6}{97} = 0.06)

Exercise 3.2.19

What is the probability of drawing a red card in a standard deck of 52 cards?

Exercise 3.2.20

What is the probability of drawing a club in a standard deck of 52 cards?

**Answer**

(frac{13}{52} = frac{1}{4} = 0.25)

Exercise 3.2.21

What is the probability of rolling an even number of dots with a fair, six-sided die numbered one through six?

Exercise 3.2.22

What is the probability of rolling a prime number of dots with a fair, six-sided die numbered one through six?

**Answer**

(frac{3}{6} = frac{1}{2} = 0.5)

*Use the following information to answer the next two exercises.* You see a game at a local fair. You have to throw a dart at a color wheel. Each section on the color wheel is equal in area.

**Figure 3.2.1.**

Let ( ext{B} =) the event of landing on blue.

Let ( ext{R} =) the event of landing on red.

Let ( ext{G} =) the event of landing on green.

Let ( ext{Y} =) the event of landing on yellow.

Exercise 3.2.23

If you land on ( ext{Y}), you get the biggest prize. Find (P( ext{Y})).

Exercise 3.2.24

If you land on red, you don’t get a prize. What is (P( ext{R}))?

**Answer**

( ext{P}(R) = frac{4}{8} = 0.5)

*Use the following information to answer the next ten exercises.* On a baseball team, there are infielders and outfielders. Some players are great hitters, and some players are not great hitters.

Let ( ext{I} =) the event that a player in an infielder.

Let ( ext{O} =) the event that a player is an outfielder.

Let ( ext{H} =) the event that a player is a great hitter.

Let ( ext{N} =) the event that a player is not a great hitter.

Exercise 3.2.25

Write the symbols for the probability that a player is not an outfielder.

Exercise 3.2.26

Write the symbols for the probability that a player is an outfielder or is a great hitter.

**Answer**

(P( ext{O OR H}))

Exercise 3.2.27

Write the symbols for the probability that a player is an infielder and is not a great hitter.

Exercise 3.2.28

Write the symbols for the probability that a player is a great hitter, given that the player is an infielder.

**Answer**

(P( ext{H|I}))

Exercise 3.2.29

Write the symbols for the probability that a player is an infielder, given that the player is a great hitter.

Exercise 3.2.30

Write the symbols for the probability that of all the outfielders, a player is not a great hitter.

**Answer**

(P( ext{N|O}))

Exercise 3.2.31

Write the symbols for the probability that of all the great hitters, a player is an outfielder.

Exercise 3.2.32

Write the symbols for the probability that a player is an infielder or is not a great hitter.

**Answer**

(P( ext{I OR N}))

Exercise 3.2.33

Write the symbols for the probability that a player is an outfielder and is a great hitter.

Exercise 3.2.34

Write the symbols for the probability that a player is an infielder.

**Answer**

(P( ext{I}))

Exercise 3.2.35

What is the word for the set of all possible outcomes?

Exercise 3.2.36

What is conditional probability?

**Answer**

The likelihood that an event will occur given that another event has already occurred.

Exercise 3.2.37

A shelf holds 12 books. Eight are fiction and the rest are nonfiction. Each is a different book with a unique title. The fiction books are numbered one to eight. The nonfiction books are numbered one to four. Randomly select one book

Let ( ext{F} =) event that book is fiction

Let ( ext{N} =) event that book is nonfiction

What is the sample space?

Exercise 3.2.38

What is the sum of the probabilities of an event and its complement?

**Answer**

1

*Use the following information to answer the next two exercises.* You are rolling a fair, six-sided number cube. Let ( ext{E} =) the event that it lands on an even number. Let ( ext{M} =) the event that it lands on a multiple of three.

Exercise 3.2.39

What does (P( ext{E|M})) mean in words?

Exercise 3.2.40

What does (P( ext{E OR M})) mean in words?

**Answer**

the probability of landing on an even number or a multiple of three

## Ken Ward's Mathematics Pages

A series is a set of numbers such as:

1+2+3

which has a sum. A series is sometimes called a progression , as in "Arithmetic Progression".

A sequence, on the other hand, is a set of numbers such as:

2,1,3

where the order of the numbers is important. A different sequence from the above is:

1, 2, 3

A series such as:

1+2+3.

has the same sum as:

2+1+3

but the numbers are in a different sequence.

## 20 Most Common Math Terms and Symbols in English

Below is a summary of the common mathematical symbols discussed below, along with the words in English used to describe them.

Math can be frustrating enough in your own language. But when learning a new language, you may find that you’ll need to relearn not just numbers, but many of the terms used in the world of math.

For example, it might be difficult for you to calculate a tip at a restaurant out loud for your English-speaking friend, but something like that can definitely come in handy. To help, here are a bunch of terms (and example equations) that English speakers use when rattling their brains with numbers and equations.

### Addition

6 + 4 = 12

Six plus four equals twelve.

This type of calculation is called **addition** , which is when you add two or more numbers together. When saying the equation out loud, we use the w or d “ plus ,” and the “ + ” symbol is called a **plus sign** . The result of an addition equation is called a **sum** .

### Equation

Usually, we say that one expression **equals** another, and the “=” symbol is fittingly called an **equals sign** . Though it is fairly common in English to say the word “equals,” it is also fine to use the singular “is.” For example, two plus three *is* five. Any mathematical statement involving an equals sign is called an **equation** .

### Not-equals sign

6 + 4 ≠ 13

Six plus four is not equal to thirteen.

The “≠” symbol is called a **not-equals sign** , and we say that one expression is **not equal to** another.

### Subtraction

15 – 8 = 7

Fifteen minus eight equals seven.

This type of calculation is called **subtraction** , which is when you **subtract** one number from the other to get a difference. When saying the equation out loud, we use the word “minus,” and the “-” symbol is called—you guessed it—a **minus sign** . However, the word “minus” is not used when describing negative numbers (as opposed to positive numbers). For example, three minus four is not “minus one,” but “ **negative** one.”

### Plus-minus sign

4 ± 3 = 1 or 7

Four plus or minus three equals one or seven.

The “±” symbol is called the **plus-minus** **sign** , and when used in an equation, we say that one number **plus or minus** another results in two possible sums.

### Multiplication

5 × 2 = 10

Five times two equals ten.

Five multiplied by two equals ten.

Now we’ve gotten to **multiplication** , and there are two ways to recite such a calculation. One way is to say that one number times another results in a product. The other way is to use the logical term “ **multiplied by** .” The “×” symbol is considered to be the **multiplication sign** , although you can also use a dot (⋅) or an asterisk (∗).

### Division

21 ÷ 7 = 3

Twenty-one divided by seven equals three.

When dealing with **division** , we say that one number is **divided by** another number to get a **quotient** . We call the “÷” symbol a **division sign** , but it is also common to use a slash (/), a symbol also used for fractions. If an answer contains a remainder, then you simply say “ **remainder** ” where the “r” is. For example, 22 ÷ 7 = 3r1 would be “twenty-two divided by seven equals three remainder one.”

### Inequality

18.5 > 18

Eighteen point five is greater than eighteen.

This type of equation is called an **inequality** , and it is usually read from left to right. So logically, the “>” symbol is called a “ **greater-than sign** ” and the “<” symbol is called a “ **less-than sign** .” You can also use the “≥” or “≤” symbols if a number, usually a variable, may be **greater than or equal to** another number, or **less than or equal to** it.

### Decimal

18.5 is considered a **decimal** , and the period used to write this number is called a **decimal point** .

When said out loud, we usually use the word “point,” followed by a string of individual numbers. For example, 3.141 would be pronounced “three point one four one.” However, with simpler numbers, it is common to use a fraction like “five-tenths.” Don’t worry, this will be covered next.

Money tends to be recited a little differently. For example, if something costs $5.75, you wouldn’t say “five point seven five dollars.” Instead you would say “five dollars and seventy-five cents” or simply “five seventy-five.”

### Approximation

π ≈ 3.14

Pi is approximately equal to 3.14

This type of equation is called an **approximation** , where one value is **approximately equal to** another value. The “≈” symbol is called an **almost-equals sign**.

The fields of math and science tend to borrow a lot of letters from the Greek alphabet as commonplace symbols, and English tends to put a twist on the pronunciation of these letters. For example, the letter π is not pronounced /pi/ as it normally would be, but rather as /paj/, like the word “pie.”

Be careful about pronouncing Greek letters in English because oftentimes, it won’t be the same.

### Ratio (numerator, denominator)

1 ÷ 3 = ⅓

One divided by three equals a third.

In a fraction, the top number is called the **numerator** and the bottom number is called the **denominator** . When saying fractions out loud, we usually treat the denominator like an ordinal number. That means ⅓ is pronounced “a third,” ¼ is pronounced “a fourth,” etc. One exception is ½, which is usually called “a **half** ,” not “a second.” Similarly, ¼ can be called “a **quarter** ,” as well as a fourth, but those are the only irregularities.

With all of these fractions, it’s acceptable to use the word “one” instead of “a,” so ½ can be called “one half” as well as “a half.” And if the numerator is a number greater than one, simply say that number out loud. ¾ would be “three-fourths,” ⅖ would be “two-fifths,” etc. Notice the use of a hyphen when writing out the fraction.

With any fraction, it is also possible to simply say that one number is “over” another. While ⅖ can be pronounced “two-fifths,” it is also perfectly fine to say “two over five.” In fact, when dealing with **variables** (letters that represent numbers), it is actually the only convenient way to say it. For example, x/y would be said as “x over y,” while nobody would ever say “x-yth.”

### Improper fraction

2 ÷ 3 = 1½

Two divided by three equals one and a half.

An **improper fraction** is a combination of a whole number ( **integer** ) and a fraction and involves the use of the word “and.” So 1½ would be one *and* a half, 2¾ would be two *and* three-fourths, etc. As stated before, decimals can occasionally be stated as an improper fraction. While it is normal to pronounce 0.7 as “zero point seven” or “point seven,” it can also be said as “seven-tenths,” since it is technically equal to 7/10. Similarly, 0.75 can be said as “seventy-five hundredths.”

However, this method of reading decimals can become clunky and confusing, and so it is much more common and convenient to stick with the “point” method.

### Percentage

20 × 40% = 8

Twenty times forty percent equals eight.

Forty percent of twenty is eight.

The **percent sign** (%) is used to indicate a **percentage** . When reading a percentage, you simply say the number and the word “ **percent** ” after it, so 50% would be read as “fifty percent.” When calculating something that involves a percentage, you can simply pronounce it as a standard multiplication equation, or you can say that a certain percent of another number results in a product.

In computer science, the percent sign tends to have a different function and is actually used as the **modulo operator** , which acts as a division calculation but outputs only the remainder. Where the percent sign is, you would say “ **modulo** ” or “ **mod** ” for short. For example, 15 % 6 == 3 would be “fifteen mod six equals three” (a double percent sign is usually used in computer languages, but it is read the same).

### Exponential

3 3 = 27

Three cubed equals twenty-seven.

Three to the third equals twenty-seven.

Three to the power of three equals twenty-seven.

An **exponent** is when you take a number and multiply it by itself a certain number of times, an operation called **exponentiation** . In other words, you take one number **to the power** **of** another number. This is the easiest way to read an exponent out loud, since it works easily with decimals and fractions (“four to the seven point five,” “three to the four-fifths,” etc.).

However, it is also common to use an ordinal number when reading aloud an exponent. For example, x 3 reads “x to the third,” x 4 reads “x to the fourth,” etc. Note that this is different from saying “x-thirds” or “x-fourths,” which would turn the number into a fraction.

It is not common to say x 2 as “x to the second.” Instead, the convention is to say “x squared,” which relates to concepts of geometry. Similarly, it is common to say x 3 as “x cubed.”

However, there is no equivalent for x 4 and numbers beyond that. “Squared” and “cubed” are also used when talking about units of length in two or three dimensions. For example, 5 ft 2 would be read as “five feet squared,” and 50 km 3 would be read as “fifty kilometers cubed.

### Square root

√16 = 4

The square root of sixteen is four.

The result of this equation is called a **square root** , and the “√” symbol is called a **radical sign** (“radical” literally means “root”). It is typical to state that the square root of one number equals another number.

A square root is essentially a number to the power of a half. In other words, √16 is the same as 16 1/2 . However, if the number is to the power of a different fraction, say ⅓, then the root becomes a **cube root** , written as 3 √16.

For this, you can say “the cube root of sixteen,” but you can also say “sixteen root three.” Similarly, 4 √16 would be “sixteen root four,” etc.

### Imaginary number

√(–4) = 2i

The square root of negative four is two i.

An **imaginary number** is the result of taking the square root of a negative number. When reading an imaginary number aloud, simply pronounce the letter “i” as it is. 2i is pronounced “two i,” 3i is “three i,” etc.

### Logarithm

log

_{2}8 = 3

Log base two of eight equals three.

A **logarithm** is basically an inverse of an exponential equation, and though it seems complicated, reading one may actually be easier and more consistent.

In the case of log_{2}8, since the “2” is considered to be the base of the logarithm, you would say that log base two of eight equals three. An expression containing “ln” is called a **natural log** . For example, lnx would be stated as “the natural log of x.”

12m / 4s = 3m/s

Twelve meters divided by four seconds equals three meters per second.

- This class will meet five times
**per**(Five times a week) - I usually assist ten customers
**per**(Ten customers every shift)

### Infinity

0 < x < ∞

X is greater than zero and less than infinity.

**Infinity** (∞) is an abstraction of the largest number imaginable, the opposite of which is **negative infinity** (–∞). The “∞” symbol is called the **infinity symbol** , sometimes called a **lemniscate** because of its figure-eight shape. Notice that it’s different from the word “infinite,” which is an adjective that describes something that is endless or limitless.

### Factorial

A **factorial** is represented by an exclamation point, and you simply say the word “factorial” after the number. Things don’t get much easier…

### Equation of those number

5 x (4 + 3) = 35

Five times the quantity of four plus three equals thirty-five.

Saying equations out loud can get a bit tricky when there are parentheses involved.

One method is to take short pauses before saying numbers grouped in parentheses. But a more effective way would be to call them **the quantity of** those numbers, almost as if you’re making a calculation within a calculation, which is essentially what you’re doing.

This phrase also comes in handy when you’re dealing with complex fractions. For example, an easy way to say x / (y + z) would be “x over the quantity of y plus z.”

## Courses and Curriculum

MATH-ACM students may double count 15 credits (5 courses) of 500 level courses toward both the B.S./A.B. in Mathematics and M.S. in Applied and Computational Mathematics degrees. The courses double counted must be elected at the 500 level and include:

- Math 551 (Advanced Calculus), Math 562 (Math Modeling), and either Math 572 (Numerical Analysis) or Math 573 (Matrix Computations). This satisfies part of both the Analysis/Algebra option and the Applied Courses for the B.S. and all of the Core requirements for the M.S. degree.
- And
**either**Option I, II or III:

Two additional electives that satisfy both the B.S. degree electives and the Modeling Specialization requirements of the M.S. degree. Choices include: Math 504, Math 520, Math 525, Math 554, Math 558, Math 523, Math 514, Math 516.

Two courses that satisfy the cognate option for the B.S. degree. Choices include: Stat 530, Stat 535, Stat 545, Stat 560, or select courses at the graduate level from CIS, ECE, ECON, IMSE, ME, PHYS and others.

One course from Option I and one course from the Option II.

*A student may not receive credit for both a 400 and 500 level equivalent courses (for example, both Math 455 and Math 555).*

## Checked Indexed Accesses ( --noUncheckedIndexedAccess )

TypeScript has a feature called *index signatures*. These signatures are a way to signal to the type system that users can access arbitrarily-named properties.

In the above example, Options has an index signature that says any accessed property that’s not already listed should have the type string | number . This is often convenient for optimistic code that assumes you know what you’re doing, but the truth is that most values in JavaScript do not support every potential property name. Most types will not, for example, have a value for a property key created by Math.random() like in the previous example. For many users, this behavior was undesirable, and felt like it wasn’t leveraging the full strict-checking of --strictNullChecks .

That’s why TypeScript 4.1 ships with a new flag called --noUncheckedIndexedAccess . Under this new mode, every property access (like foo.bar ) or indexed access (like foo["bar"] ) that ends up resolving to an index signature is considered potentially undefined. That means that in our last example, opts.yadda will have the type string | number | undefined as opposed to just string | number . If you need to access that property, you’ll either have to check for its existence first or use a non-null assertion operator (the postfix ! character).

One consequence of using --noUncheckedIndexedAccess is that indexing into an array is also more strictly checked, even in a bounds-checked loop.

If you don’t need the indexes, you can iterate over individual elements by using a for – of loop or a forEach call.

This flag can be handy for catching out-of-bounds errors, but it might be noisy for a lot of code, so it is not automatically enabled by the --strict flag however, if this feature is interesting to you, you should feel free to try it and determine whether it makes sense for your team’s codebase!

## English for Maths

Maths students **of English 2** will not sit June 2021 exams. Instead, they will be evaluated by making an o**ral presentation through Webex – with cameras on – on a science-oriented** **topic.** (You may choose from a wide range of areas of interest. Your topic should be **relevant to your School of Technology & Engineering**, while it can be possibly combined with topics taken from schools of **different sciences, or even arts, as well.)** The presentation should strictly last **10 minutes**. Your first 5 minutes should focus on the presentation of your topic. The last 5 minutes should be based on a brief oral commentary of all elements of academic language that you have used in an essay (without a word limit) written by you on the same topic (**coherence and cohesion devices, types of sentences, features of academic writing, topic sentence, controlling idea, primary and secondary supports, conclusion, method of development, pattern, sentence structure and sentence style)**. These elements **should be demonstrated on the screen** (you decide on your highlighting system, i.e., using different colours, initial letters, notes etc.). Helpful **guidelines and links** with respect to a successful presentation **are provided through the blog**. Try to be **fluent, communicative and confident**. **Never just read** your written work. Show that you are **familiar** enough with your topic and linguistic analysis. You should also **refer to your sources and bibliography** while applying the **strategies of summarizing and paraphrasing**, thus **avoiding plagiarism**. Try to produce your own written texts. ** Copying somebody else’s original work will be considered a fail.**On your presentation day you are required to

**submit an electronic copy of your work.**

__Your mark will be based on your oral performance of content and language.__Presentations will take place__in the first two weeks after Easter vacations__. As for the exact dates of your presentation__you will be informed in one of these days.__You will be sent the available days and times__through google docs__, so that you could__sign up.__The__same system__will take place in__September 2021 exams__.Maria Koutraki ([email protected] [email protected])

## Common Core Math Vocabulary & Standards

The Math Common Core State Standards provide clear goals defining what students should understand and be able to do at every grade level. On every math page, there is a “standards overview table” summarizing the Common Core Standards’ math learning goals and skills for that grade and content area.

The Common Core State Standards for Mathematical Practice establish eight main math skills that K12 educators should develop in their students:

**Make sense of problems and persevere in solving them.****Reason abstractly and quantitatively.****Construct viable arguments and critique the reasoning of others.****Model with mathematics.****Use appropriate tools strategically.****Attend to precision.****Look for and make use of structure.****Look for and express regularity in repeated reasoning.**

## Parts of an Expression

Algebraic expressions are combinations of variables , numbers, and at least one arithmetic operation.

For example, 2 x + 4 y &minus 9 is an algebraic expression.

**Term:** Each expression is made up of terms. A term can be a signed number, a variable, or a constant multiplied by a variable or variables.

**Factor:** Something which is multiplied by something else. A factor can be a number, variable, term, or a longer expression. For example, the expression 7 x ( y + 3 ) has three factors: 7 , x , and ( y + 3 ) .

**Coefficient:** The numerical factor of a multiplication expression that contains a variable. Consider the expression in the figure above, 2 x + 4 y &minus 9 . In the first term, 2 x , the coefficient is 2 : in the second term, 4 y , the coefficient is 4 .

**Constant:** A number that cannot change its value. In the expression 2 x + 4 y &minus 9 , the term 9 is a constant.

**Like Terms:** Terms that contain the same variables such as 2 m , 6 m or 3 x y and 7 x y . If an expression has more than one constant terms, those are also like terms.

Difference of a number and 7

Identify the terms, like terms, coefficients, and constants in the expression.

First, we can rewrite the subtractions as additions.

9 m &minus 5 n + 2 + m &minus 7 = 9 m + ( &minus 5 n ) + 2 + m + ( &minus 7 )

So, the **terms** are 9 m , ( &minus 5 n ) , m , 2 , and ( &minus 7 ) .

**Like terms** are terms that contain the same variables.

9 m and 9 m are a pair of **like terms** . The constant terms 2 and &minus 7 are also like terms.

**Coefficients** are the numerical parts of a term that contains a variable.

So, here the **coefficients** are 9 , ( &minus 5 ) , and 1 . ( 1 is the coefficient of the term m .)

The **constant** terms are the terms with no variables, in this case 2 and &minus 7 .

Algebraic expressions must be written and interpreted carefully. The algebraic expression 5 ( x + 9 ) is **not** equivalent to the algebraic expression, 5 x + 9 .

See the difference between the two expressions in the table below.

In writing expressions for unknown quantities, we often use standard formulas. For example, the algebraic expression for "the distance if the rate is 50 miles per hour and the time is T hours" is D = 50 T (using the formula D = R T ).

An expression like x n is called a power. Here x is the base, and n is the exponent. The exponent is the number of times the base is used as a factor. The word phrase for this expression is " x to the n th power."