Probability is the measure of how likely a particular event will occur, but figuring probability is a bit more scientific than telling fortunes and reading tarot cards. You express probability as a percentage, fraction, or decimal.

The probability of an event’s occurring falls between 0 percent and 100 percent or between 0 and 1. If the probability of an event’s occurrence is 0, or 0 percent, it’s impossible for the event to occur. If the probability is 1, or 100 percent, the event is certain to occur. Probability questions may ask you to determine the probability of one event or multiple events. 

Probability of One Event

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}.

 probability

Basic Examples

Example 1: 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 2: 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 3: 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 4: 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 5: 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. 

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)

Probability of Many Events

You can find the probability of multiple events by following several rules.

1. Special rule of addition

The probability of the occurrence of either of two possible events that are mutually exclusive. You use the special rule of addition to figure out the probability of rolling a die and coming up with either a 1 or a 2. You can’t get both on one roll, so the events are mutually exclusive. Therefore, the probability of rolling a 1 or a 2 in one roll is P(A) + P(B):

P(A or B) = 1/6 + 1/6 = 1/3

2. General rule of addition

The probability of the occurrence of either of two possible events that can happen together. For example, the probability of drawing a playing card that displays a club or a queen.

P(A or B) = P(A) + P(B) – P(A and B)

3. Special rule of multiplication

The probability of the occurrence of two events at the same time when the two events are independent of each other. The probability of multiple events occurring together is the product of the probabilities of the events occurring individually. For example, if you are rolling two dice at the same time, the probability of rolling a 1 on one die and a 2 on the other is:

P(A and B) = 1/6 × 1/6 = 1/36

4. General rule of multiplication

The probability of the occurrence of two events when the occurrence of the first event affects the outcome of the second event. Suppose the outcome of the second situation depends on the outcome of the first event. You then invoke the general rule of multiplication. The term P(B|A) is a conditional probability, where the likelihood of the second event depends on the fact that A has already occurred.

For example, to find the odds of drawing the ace of spades from a deck of 52 cards on one try and then drawing the king of spades on the second try - with the ace out of the deck - apply the formula, like this:

P(A and B) = P(A) × P(B|A)