Probability | Arithmetic & algebra | Quantitative reasoning

A probability describes how likely it is that an event will occur. You’ll often see probabilities written as percentages from 0% to 100%, which correspond to decimal values from 0 to 1.

If the event will never occur, the probability is $0$ , i.e. a $0%$ chance.
If the event will always occur, the probability is $1$ , i.e. a $100%$ chance.
If the event will occur 7 out of 10 times, the probability is $0.7$ , i.e. a $70%$ chance.

Because an event either happens or it doesn’t, these two probabilities always add up to $1$ (a $100%$ total chance). For example, suppose you pick a marble from a bag and there’s a $20%$ chance it will be blue. What’s the probability it won’t be blue? That probability is $100% - 20% = 80%$ .

You can often find a probability by dividing the number of desired outcomes by the number of total possible outcomes. For example, suppose you and your brother are both in a raffle, and you’d be happy if either of you wins. If there are 20 total people participating, then the chance that one of you wins is $2/20 = 0.1$ or $10%$ . (There are 2 desired tickets and 20 total possible tickets.)

There are two rules that are especially useful when working with probabilities: AND and OR.

AND means multiply the probabilities

Consider this question:

What is the chance of winning a coin flip twice in a row?

Coin flips are random, and the chance of winning a single coin flip is $50%$ , which is $0.50$ in decimal form. Because you’re trying to win the first flip and the second flip, this is an AND situation.

$.50 \times .50 = .25$

So, the probability of winning twice in a row is $25%$ .

Let’s try another example.

You are tasked with randomly selecting criminals from a police lineup. There are two accomplices in the crime. What is the chance you overlook both criminals when randomly selecting two from a lineup that includes 7 suspects? Express the probability in terms of a fraction.

Try to solve it yourself, then check below!

(spoiler)

Answer: $10/21$

This question asks for the probability of selecting a non-criminal and then selecting a second non-criminal, so you multiply the two probabilities.

There are $2$ criminals in a group of $7$ suspects, so there are $5$ non-criminals. The probability of selecting a non-criminal on the first pick is $5/7$ . After that pick, one non-criminal is no longer available, so the probability of selecting a non-criminal on the second pick is $4/6$ .

$5/7 \times 4/6 = 20/42 = 10/21$

OR means add the probabilities

Now consider this question:

What is the chance of randomly selecting a consonant or the letter A from the alphabet?

There are 21 consonants in the alphabet, and there is 1 letter A. Because this is an OR situation, you add the probabilities. There is a $21/26$ chance of selecting a consonant and a $1/26$ chance of selecting the letter A. So the total probability of selecting either a consonant or A is:

$21/26 + 1/26 = 22/26 = 11/13$

As a decimal, $11/13$ is approximately $85%$ .

Let’s try another example.

Probabilities $p$ , $q$ , and $r$ are the chances of three mutually exclusive events occurring. No other possible events can occur beyond these three events.

Quantity A: The chance that $p$ or $r$ occurs
Quantity B: $1 - q$

Give it a try!

(spoiler)

Answer: C. The quantities are equal

The probability that $p$ or $r$ occurs is $p + r$ .

In Quantity B, the $1$ represents all possible outcomes (a 100% total chance). Since exactly one of the three mutually exclusive events must occur, you can write:

$1 = p + q + r$

Rearranging gives Quantity B:

$1 1 - q = p + q + r = p + r$

So both Quantity A and Quantity B equal $p + r$ .

Mutually exclusive and independent events

You just saw a question involving independent events and another involving mutually exclusive events. It’s important to keep these categories straight.

If two events are mutually exclusive, they cannot occur at the same time. For example, if it is snowing outside, it cannot be 100 degrees F. These events are mutually exclusive because snow requires temperatures below freezing.

Independent events, however, do not affect each other. For example, whether it’s raining does not affect whether a random moment falls on a weekday or weekend. The probability that a random moment is on the weekend is always $2/7$ , whether it is raining or not.

Although this isn’t a perfect rule, you’ll often:

add probabilities for mutually exclusive events
multiply probabilities for independent events

This matches what you did in the practice problems above: you added mutually exclusive events and multiplied independent events.

Probability notation

On the GRE, probabilities are often written using notation like $P (B) = .70$ . Here, $P$ means “probability,” and the letter in parentheses names the event. Any probability must be a number between 0 and 1, so $P (B) = .70$ means there is a 70% chance that event $B$ will occur.

For mutually exclusive events, the notation $P (A or B)$ refers to the sum of $P (A)$ and $P (B)$ . For independent events, the notation $P (A and B)$ refers to the product of $P (A)$ and $P (B)$ .

In the Venn diagram and tables section of this course, you learned about the union and intersection of sets. The same ideas apply to probabilities:

The union of two events is the probability that one or both events occur.
The intersection of two events is the probability that both events occur.

Be careful: the union is not the probability that exactly one of the events occurs. “Union” means either event could occur, including the possibility that both occur.

The notation for the union and intersection of probabilities, along with a visual representation of each, is shown below.

Notation for the union and intersection of probabilities 1

Notation for the union and intersection of probabilities 2

Notation for the union and intersection of probabilities 3

Common themes

A probability can be described as a fraction, decimal, or percent.
A probability can never be less than $0$ or greater than $1$ .
The chance of an event NOT occurring is $1$ minus that probability.
Probability questions often involve permutation and combinations, where the total combinations or permutations is the denominator and the desired permutations or combinations is the numerator. This is consistent with our understanding that a probability is the desired outcomes over the total possible outcomes.

A probability describes how likely it is that an event will occur. You’ll often see probabilities written as percentages from 0% to 100%, which correspond to decimal values from 0 to 1.

If the event will never occur, the probability is $0$ , i.e. a $0%$ chance.
If the event will always occur, the probability is $1$ , i.e. a $100%$ chance.
If the event will occur 7 out of 10 times, the probability is $0.7$ , i.e. a $70%$ chance.

There are two rules that are especially useful when working with probabilities: AND and OR.

AND means multiply the probabilities

Consider this question:

What is the chance of winning a coin flip twice in a row?

$.50 \times .50 = .25$

So, the probability of winning twice in a row is $25%$ .

Let’s try another example.

You are tasked with randomly selecting criminals from a police lineup. There are two accomplices in the crime. What is the chance you overlook both criminals when randomly selecting two from a lineup that includes 7 suspects? Express the probability in terms of a fraction.

Try to solve it yourself, then check below!

(spoiler)

Answer: $10/21$

This question asks for the probability of selecting a non-criminal and then selecting a second non-criminal, so you multiply the two probabilities.

$5/7 \times 4/6 = 20/42 = 10/21$

OR means add the probabilities

Now consider this question:

What is the chance of randomly selecting a consonant or the letter A from the alphabet?

$21/26 + 1/26 = 22/26 = 11/13$

As a decimal, $11/13$ is approximately $85%$ .

Let’s try another example.

Probabilities $p$ , $q$ , and $r$ are the chances of three mutually exclusive events occurring. No other possible events can occur beyond these three events.

Quantity A: The chance that $p$ or $r$ occurs
Quantity B: $1 - q$

Give it a try!

(spoiler)

Answer: C. The quantities are equal

The probability that $p$ or $r$ occurs is $p + r$ .

In Quantity B, the $1$ represents all possible outcomes (a 100% total chance). Since exactly one of the three mutually exclusive events must occur, you can write:

$1 = p + q + r$

Rearranging gives Quantity B:

$1 1 - q = p + q + r = p + r$

So both Quantity A and Quantity B equal $p + r$ .

Mutually exclusive and independent events

You just saw a question involving independent events and another involving mutually exclusive events. It’s important to keep these categories straight.

Although this isn’t a perfect rule, you’ll often:

add probabilities for mutually exclusive events
multiply probabilities for independent events

This matches what you did in the practice problems above: you added mutually exclusive events and multiplied independent events.

Probability notation

For mutually exclusive events, the notation $P (A or B)$ refers to the sum of $P (A)$ and $P (B)$ . For independent events, the notation $P (A and B)$ refers to the product of $P (A)$ and $P (B)$ .

In the Venn diagram and tables section of this course, you learned about the union and intersection of sets. The same ideas apply to probabilities:

The union of two events is the probability that one or both events occur.
The intersection of two events is the probability that both events occur.

Be careful: the union is not the probability that exactly one of the events occurs. “Union” means either event could occur, including the possibility that both occur.

The notation for the union and intersection of probabilities, along with a visual representation of each, is shown below.

Notation for the union and intersection of probabilities 1

Notation for the union and intersection of probabilities 2

Notation for the union and intersection of probabilities 3

Common themes

A probability can be described as a fraction, decimal, or percent.
A probability can never be less than $0$ or greater than $1$ .
The chance of an event NOT occurring is $1$ minus that probability.
Probability questions often involve permutation and combinations, where the total combinations or permutations is the denominator and the desired permutations or combinations is the numerator. This is consistent with our understanding that a probability is the desired outcomes over the total possible outcomes.