Textbook

A probability describes how likely it is that an event may occur. You’ll see probabilities represented as percentages from 0% to 100%, which correspond to their 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

Since something will either happen or it won’t, the sum of these two probabilities will *always* add up to $1$, i.e. a $100%$ chance. For example, imagine we’re picking a marble from a bag, and there’s a $20%$ chance it will be blue. What is the probability it won’t be blue? That probability is $100%−20%=80%$.

A probability can be found by dividing the number of desired outcomes by the number of total possible outcomes. For example, if you and your brother are participating in a raffle, and you both would be happy with either of you winning, and there are 20 total people participating in the raffle, the chance of winning the raffle is 2/20 = .1 or 10%. There are two desired tickets to be drawn and 20 total tickets possible.

There are two rules that can be helpful when working with probabilities: *AND* and *OR*.

Let’s think about this question for a moment:

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

Coin flips are luck-based, and the chance of winning a single coin flip is $50%$, represented in decimal form as $0.50$. Because we are trying to win twice in a row (i.e. winning the first flip *and* the second flip), this is an *AND* situation.

$.50×.50=.25$

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 we have to multiply the two probabilities together.

There are $2$ criminals, so given our group has $7$ suspects, there must be $5$ non-criminals. The probability of selecting a non-criminal for the first pick is $5/7$. After making this choice that one non-criminal is no longer in the lineup, so the chance of selecting a second non-criminal is $4/6$.

$5/7×4/6=20/42=10/21$

Think about 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 obviously, there is a single letter A. Because this is an *OR* situation, you should add the probabilities. There is a $21/26$ chance of selecting a consonant and a $1/26$ chance of selecting the letter A. The total probability of selecting either a consonant or A is:

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

Divided out, $11/13$ 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 $r$ could occur is equivalent to $p+r$.

In Quantity B, the $1$ represents all possible outcomes, i.e., there is a 100% chance that one of the three mutually exclusive events occurs. This can be represented as the equation $1=p+q+r$, which we can rearrange to get Quantity B’s value of $1−q$.

$11−q =p+q+r=p+r $

So, it turns out that both Quantity A and Quantity B are $p+r$.

You just saw a question that used independent events and another that referenced mutually exclusive events. It’s important to be keenly aware of the difference between these two categories of events.

If two events are mutually exclusive, they cannot occur simultaneously. For example, if it is snowing outside, it cannot be 100 degrees F, and if it is 100 degrees F, it cannot be snowing. These events are mutually exclusive because the temperature must be below freezing for it to be snowing.

Independent events, however, have no relation to each other. The chance of it raining at any given moment does not affect whether it is a weekday or weekend. No matter what, the probability that any random moment is on the weekend is 2/7, whether it is raining or not.

Though this is certainly not a hard and fast rule, you will most likely add the probabilities of mutually exclusive events and multiply the probabilities of independent events. Again, this is certainly not always the case, but the properties of these event types lend themselves to these specific operations. For example, in the two practice problems above, we added the mutually exclusive events and multiplied the independent events.

You will also see that probabilities are often described by a specific notation on the GRE. Don’t worry; it is very straightforward. Instead of saying, “The probability of event B is 70%”, a question may simply show P(B)=.70. P simply means probability, and the letter in the parentheses refers to the event. Any probability must be described as a number between 0 and 1. Therefore, if P(B)=.70, it means that there is a 70% chance that event B will occur.

For mutually exclusive events, the notation P(A or B) just refers to the sum of P(A) and P(B). For independent events, the notation P(A and B) just 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. This notation also applies to probabilities. For example, the union of two events is the probability that either occurs, and the intersection is the probability that both events would occur. Note how the union of two events is NOT the probability of just one of the two events occurring. The word “union” means that one or both of the events could occur. The notation for the union and intersection of probabilities and a visual representation for each are shown below.

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