If the occurrence of an event does not affect the occurrence of another event, then those events are called independent events. If $A$ and $B$ are independent events, then the following rule applies for the probability of this type of compound event.

Probability of $A$ and $B$
If $A$ and $B$ are independent events, then $P (A and B) = P (A) \cdot P (B) .$

We are told that

A

and

B

are independent events, that

P (A) = 50 %,

and that

P (B) = 25 % .

With this information we want to find the value of

P (A and B) .

P (A and B) = P (A) \cdot P (B)

SubstituteII

$P (A) = 50 %$ , $P (B) = 25 %$

P (A and B) = 50 % \cdot 25 %

▼

Simplify right-hand side

PercentToFrac

$a % = \frac{a}{1 0 0}$

P (A and B) = \frac{5 0}{1 0 0} \cdot \frac{2 5}{1 0 0}

MultFrac

Multiply fractions

P (A and B) = \frac{1 2 5 0}{1 0 0 0 0}

CalcQuot

Calculate quotient

P (A and B) = 0.125

WritePercent

Convert to percent

P (A and B) = 12.5 %

Extra

Compound Event

A compound event is an event that is made up of two or more events. In our exercise, the events $A$ and $B$ were independent. To find the probability that two independent events will occur, we can multiply the probabilities of each event, as we did in our solution.

Probability of $A$ and $B$
If $A$ and $B$ are independent events, then $P (A and B) = P (A) \cdot P (B) .$

Events that cannot happen at the same time are called mutually exclusive events. The following rule tells us about the probability of mutually exclusive events.

Probability of Mutually Exclusive Events
If $A$ and $B$ are mutually exclusive events, then $P (A and B) = 0,$ and $P (A or B) = P (A) + P (B) .$

Sometimes, the events can be overlapping. Overlapping events have outcomes in common. Recall the rule that tells us about the probability of such events.

Probability of Overlapping Events
If $A$ and $B$ are overlapping exclusive events, then $P (A or B) = P (A) + P (B) - P (A and B) .$

Exercise