Compound events

A compound event is an event that includes two or more simple events. Simple events are events that can have only one outcome, while compound events can have multiple different outcomes. Compound events can be made up of a number of independent events (events in which the outcome of one event has no effect on the probability of the other) or dependent events (events in which the outcome of one event affects the probability of another).

The concept of compound events is used for determining compound probabilities.

Compound probability

A compound probability is the probability of a compound event. Generally, it is the ratio of favorable outcomes to the total number of outcomes within the sample space of the compound event and can be calculated using one of two rules: the addition rule and the multiplication rule.

Addition rule

The addition rule can be used for compound events in which the simple events involved do not occur together. This is referred to as being "mutually exclusive." For example, a person cannot weigh 160 pounds and 162 pounds at the same time. They can only be one or the other, so the probability of someone being 160 pounds or 162 pounds is mutually exclusive.

To find the probability that any one of several mutually exclusive events occurs, use the addition rule, and add the probabilities of each event:

P(A or B) = P(A) + P(B)

Example

Reference the table below, and find the probability that a randomly selected student will weigh 50 kilograms (~130 pounds) or more.

Probability distribution for the
weight of students in a class
Weight (kg) Relative frequency
54 or more 0.09
53 0.07
52 0.09
51 0.14
50 0.17
49 0.15
48 0.08
47 0.06
46 0.04
45 or less 0.11
   Total: 1.00

Given that X is the random variable representing a student's weight, the probability that a student weighs 50 kg or more can be written as:

P(X ≥ 50)

Since the weight of a student is a mutually exclusive event, the probability of a randomly selected student weighing 50 kg or more can be found using the addition rule by summing the probabilities of all the weights 50 kg or greater:

P(X ≥ 50) = P(50) + P(51) + P(52) + P(53) + P(54 or more)

P(X ≥ 50) = 0.17 + 0.14 + 0.09 + 0.07 + 0.09 = 0.56

There is a 56% chance of a randomly selected student weighing 50 kg or more.

For events that aren't mutually exclusive, the addition rule can still be used, but overlap between the various outcomes needs to be taken into account. For example, assume that students are male with a probability of 0.55, taller than 5'4" with a probability of 0.35, or both with a probability of 0.10. To find the probability that a student is either male or taller than 5'4", we can add the first two probabilities (0.55 + 0.35 = 0.90), but need to subtract the probability that they are both, otherwise these students would be counted twice. Given that P(A) is the probability that a student is male, and P(B) is the probability that the student is taller than 5'4":

P(A or B) = P(A) + P(B) - P(A and B)

Therefore, the probability that a student is either male or taller than 5'4" is:

0.55 + 0.35 - 0.10 = 0.80

Multiplication rule

The multiplication rule can be used to determine the probability of a cluster of simple events depending on whether the events are independent events or dependent events.

Probability of independent events

The probability of a compound event where the events are independent events can be found by multiplying the probabilities of each independent event that makes up the compound event. Given two events, A and B, with probabilities of P(A) and P(B), the probability of both events occurring is:

P(A and B) = P(A)·P(B)

Example

What is the probability of a die that is rolled twice landing on 3 both times?

Each time the die is rolled, it constitutes an independent event, so the outcome of the roll of a die does not affect the outcome of subsequent rolls. Given that P(A) is the probability of the first roll landing on 3, and P(B) is the probability of the second roll landing on 3:

P(A and B) = P(A) · P(B) = 1/6 × 1/6 = 1/36 = 0.278

There is approximately a 27.8% chance of a fair die landing on 3 both times in 2 rolls.

Probability of dependent events

The probability of a compound event where the events are dependent events can be found by first calculating the probability of the first event, then calculating the probability of the second event occurring given that the first has already occurred (the conditional probability of the second event given the first). Multiplying the probability of the first event by the conditional probability of the second event, given the first, results in the probability of both events occurring:

P(A and B) = P(A)·P(B|A)

As an example, assume that 20% of the students in a high school are seniors and that 40% of students in the high school have taken pre-calculus. If being a senior had no effect on whether or not a student had taken pre-calculus, then the probability of being a senior and having taken pre-calculus would be (0.20)·(0.40) = 0.08. However, if it is observed that being a senior increases the probability that a student has taken pre-calculus to 65%, the above probability would be incorrect. The 40% chance of a high school student having taken pre-calculus would need to be adjusted to take into account that a senior is more likely to have taken the course.

The probability that a student is both a senior and has taken pre-calculus is therefore the probability of a student being a senior multiplied by the probability of the student having taken pre-calculus given that they are a senior:

P(senior and pre-calculus) = P(senior)·P(pre-calculus|senior)

P(senior and pre-calculus) = (0.20)·(0.65) = 0.13

Therefore, there is actually a 13% chance that a student is both a senior and has taken pre-calculus, rather than an 8% chance, since the latter doesn't take into account the increased probability of having taken pre-calculus by the time the student is a senior.