### Definition of Probability

Probability is concerned with experiments that have a finite number of outcomes. Given such an experiment, an event is a particular set of outcomes. For example, rolling a number cube with faces numbered 1 to 6 (similar to a 6-sided die) is an experiment with 6 possible outcomes: 1, 2, 3, 4, 5, or 6. One event in this experiment is that the outcome is 4, denoted as {4}; another event is that the outcome is an odd number: {1, 3, 5}.

### Basic Examples

**Example - Flipping a coin:** What’s the probability of getting heads when flipping a coin?

There is only one way to get heads in a coin toss. Hence, the top of the probability fraction is 1. There are two possible results: heads or tails. Forming the probability fraction gives 1/2.

**Example - Tossing a die:** What’s the probability of getting a 3 when tossing a die?

A die (a cube) has six faces, numbered 1 through 6. There is only one way to get a 3. Hence, the top of the fraction is 1. There are 6 possible results: 1, 2, 3, 4, 5, and 6. Forming the probability fraction gives 1/6.

**Example - Drawing a card from a deck:** What’s the probability of getting a king when drawing a card from a deck of cards?

A deck of cards has four kings, so there are 4 ways to get a king. Hence, the top of the fraction is 4. There are 52 total cards in a deck. Forming the probability fraction gives 4/52, which reduces to 1/13. Hence, there is 1 chance in 13 of getting a king.

**Example - Drawing marbles from a bowl:** What’s the probability of drawing a blue marble from a bowl containing 4 red marbles, 5 blue marbles, and 5 green marbles?

There are five ways of drawing a blue marble. Hence, the top of the fraction is 5. There are 14 (= 4 + 5 + 5) possible results. Forming the probability fraction gives 5/14.

**Example - Drawing marbles from a bowl (second drawing):** What’s the probability of drawing a red marble from the same bowl, given that the first marble drawn was blue and was not placed back in the bowl?

There are four ways of drawing a red marble. Hence, the top of the fraction is 4. Since the blue marble from the first drawing was not replaced, there are only 4 blue marbles remaining. Hence, there are 13 (= 4 + 4 + 5) possible results. Forming the probability fraction gives 4/13.

### GMAT Probability Rules

Rule No.1: AND means that you have to multiply

Rule No.2: OR means you have to add

**P(A OR B) = P(A) + P(B) – P(A AND B)**

### Disjoint or Mutually Exclusive Events

Two events are disjoint if they are mutually exclusive. Two events are disjoint if the probability of their simultaneous occurrence is zero. It is absolutely impossible to have them both happen at the same time. For example, while tossing a coin it is not possible to get both heads and tail simultaneously (Note the word *and*). Thus for mutually exclusive or disjoint events,

P(A AND B) = 0

P(A OR B) = P(A) + P(B)