GRE Exam: A Simple Guide To Solving Probability Problems

Probability refers to the likelihood that a given event will occur.

To calculate the probability of a single event, use the following formula:

The formula is illustrated in the following example.

A certain jar has 3 red marbles, 5 black marbles, and 7 white marbles. If a marble is to be selected at random from the jar, what is the probability that a black marble will be selected?






SOLUTION: In the example, an “outcome” is considered the event of selecting a marble. Since there are 15 marbles to select, there are 15 total outcomes. The “desired outcomes” refers to the number of events that satisfy the outcome you want. Since there are 5 black marbles, there are five “desired outcomes.”

Using the formula, the probability of selecting a black marble is thus: desired outcomes/total outcomes .

Mutually Exclusive Events
Imagine hearing that there is a 0.6 chance of rain. What would be the chance that it will not rain? 0.4. Why? Because the two events are mutually exclusive. Either it will rain or it won’t, so the probability of the events must add up to 1.

This example highlights the following rule: when events are mutually exclusive, the sum of their probabilities is 1.
For this question, write your answer in the box.
A certain event has two possible outcomes, a and b. If the probability of a is , and the probability of b is , what is the value of p?


SOLUTION: Since the event has only two outcomes, the probability of a and the probability of b must add to 1. Thus . To solve for p, multiply across the equation by 15:


Probability of Multiple Events
In the previous example, you looked at the probability of a single event occurring (selecting a black marble). Tougher probability questions will concern the probability that multiple events will occur.

These questions will take two forms: probability questions with “and” and probability questions with “or.”

Probability with “or”: Add Them Up
If a probability question concerns the likelihood that one event or another event will happen, add up the probabilities of each event.

For example:
In a certain bookcase, 10 books are about science, 8 are about literature, 5 are about history, and 2 are about psychology. If a book is selected at random, what is the probability that the book is about science or history?
0.08
0.2
0.3
0.4
0.6

SOLUTION: Two different events, a science book or history book, would satisfy the desired outcome. Thus to determine the number of desired outcomes, you should add the number of science books and the number of history books: 10 + 5 = 15. The total number of outcomes is the sum of the books: 10 + 8 + 5 + 2 = 25. The probability of selecting a science book or history book is thus . The correct answer is E.
 

Probability with “and”: Multiply Them
If a question concerns the probability that an event occur multiple times, multiply the probabilities of each event.

Example: If a fair-sided coin is flipped three times, what is the probability the coin will land on heads all three times?
0.125
0.25
0.5
0.75
1.5

SOLUTION: On any given flip, the probability that the coin will land on heads is 0.5. Since the question concerns the probability that the coin will land on heads all three times it is flipped, you should multiply these probabilities. 0.5 × 0.5 × 0.5 = 0.125. The correct answer is A.

 

Counting

Fundamental Counting Principle
To calculate the number of ways to combine groups, calculate the product of the number of elements in each group.

Example: Bob’s wardrobe contains 5 shirts, 7 pairs of pants, and 4 ties. If he wants to select an outfit that consists of 1 shirt, 1 pair of pants, and 1 tie, how many different outfits can he select?
16
20
28
35
140

SOLUTION: Since the question concerns the combination of different groups, find the product of the number of elements in each group. 5 × 7 × 4 = 140. The correct answer is E.

Permutations
One application of the fundamental counting principle concerns questions that ask you for the number of ways the items in a set can be ordered. These are called permutation questions and usually use the words orderings or arrangements.

To answer such questions, you should use the slot method.

Let’s use the following example to illustrate how the slot method works:

For this question, write your answer in the box.
How many five-digit locker codes can be created from the digits 0–9, inclusive, if no digit can repeat?


SOLUTION: The order of the digits is relevant to the question, so consider it a permutation and use the slot method.
Step 1: Set up a number of slots corresponding to the number of items that are being selected. Since the code is five digits, you will set up five slots:

Step 2: Starting with the first slot, put in the number of possible choices for each slot. Since you are selecting from 10 digits, there are 10 possibilities for the first slot. Since the digits cannot repeat, whichever number occupies the first slot cannot occupy the second slot. Thus there are 9 possibilities for the second slot. By the same reasoning, there are 8 possibilities for the third slot, 7 possibilities for the fourth slot, and 6 possibilities for the fifth slot.

The slots should thus look like the following:

Step 3: Multiply across.