# Probabilities

The GRE frequently tests probability, which is the odds of a particular condition occurring.

### Rules

The probability of an event happening MORE THAN ONCE is equal to:

(odds the 1st time)(odds the second 2nd)(etc.)

The probability of an event happening AT LEAST ONCE over many tries is:

1 - (odds of failure the first time)(odds of failure the second time)(etc.)

Probability is the likelihood of an event occurring. It is generally calculated as the number of desired outcomes desired by the total number of outcomes. Probability is sometimes expressed as a fraction, or as a decimal. When expressed as a decimal, a probability of “1” equals 100% chance, while a probability of .5 equals a 1 in 2 chance.

For example, when rolling a 6-sided die with faces numbered 1 to 6, the probability of rolling a 6 is 1 in 6, or 1.6666 repeating. (Assuming that the die is not imbalanced).

For the odds of rolling two 6s in a row on a 6-sided die, the formula is (odds the first time)(odds the second time).

The odds of rolling a six at least once over 2 tries on a 6-sided die. The formula is 1 - (combined odds of failure).

### Practice

1. Brian is sending out job applications to law firms. His odds of being hired at Firm A are 1 in 3; his odds at Firm B are 1 in 5, and his odds at Firm C are 1 in 7. What are the odds that Brian is hired by at least one of the law firms?

2. Melanie is shooting baskets at the gym. She has a 1 in 3 chance of making a three-point shot. What are the odds of Melanie making 2 shots in a row?

3. Darren has a 90% chance of being late for class each day. What are the odds that he shows up on time three times in a row?

The distinction between the odds of something happening "every time" versus "at least once" is sometimes known as "straight" versus "cumulative" probability.

Probability can be one of the most complex areas of mathematics; however, GRE probability is usually relatively simple. The questions the GRE asks with regard to probability can almost always be solved with the above equations.

1. 57 out of 105

Since this is the odds of something happening at least once over multiple tries, the formula is 1 - (odds of failure the first time)(odds of failure the second time)(etc.).

The odds of success for Brian's applications are $\tfrac{1}{3}$, $\tfrac{1}{5}$ and $\tfrac{1}{7}$. Thus, the odds of failure for Brian's applications are $\tfrac{2}{3}$, $\tfrac{4}{5}$, and $\tfrac{6}{7}$.

Brian's odds of being hired by at lest one of the firms are thus (1 - odds of failure), which works out to (1 - $\tfrac{2}{3}$($\tfrac{4}{5}$)($\tfrac{6}{7}$) or $\tfrac{57}{105}$

2. 1 out of 9

Since this is the odds of something happening more than once in a row, the formula is (odds the first time)(odds the second time)(etc.). The odds are: $\tfrac{1}{3}$($\tfrac{1}{3}$)

= $\tfrac{1}{9}$

3. 1 in 1000

Since this is the odds of something happening more than once in a row, the formula is (odds the first time)(odds the second time)(etc.). The odds are: $\tfrac{1}{10}$($\tfrac{1}{10}$)($\tfrac{1}{10}$)

= $\tfrac{1}{1000}$