The Multiplication Rule
The multiplication rule is used to calculate the probability of one event and another event occurring.
Below is the general equation used to calculate the probability of both events occurring by multiplying their probabilities.
P(A and B) = P(A) * P(B|A)
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The vertical bar in P(B|A) means "given," so this could also be read as "the probability that B occurs given that A has occurred."
There are some considerations to make when using this rule. Hover over the boxes below to understand how this rule applies to different events :
Independent Events
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If the events are independent, the first event happening does not impact the probability of the second event.
In the case below, the events are independent. Getting a heads in a coin flip the first time does not effect the probability of getting a heads a second time. So, to answer the question: what is the probability of flipping a fair coin and getting "heads" twice in a row:
P(one heads) = ½
P (both heads)= P (first heads) * P(second heads) = ½ * ½ = ¼
Therefore, P(both heads) = ¼
Mutually Exclusive Events
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If the events are mutually exclusive, the first event happening impacts the probability of the second event.
When events are mutually exclusive, this means that the probability of an event happening is dependent upon conditional probability (the probability of an event happening given that another event has already happened) as the second event relies on the first event.
For example, there are 3 labradors and 2 german shepherds at a pet adoption center. If a person wants to adopt two dogs randomly, what is the probability of adopting two labradors?
So the probability of adopting the first labrador is 3 out of 5 (â…—) as there 3 labradors out of a total of 5 dogs.
Now, the probability of adopting a second labrador is going to be 2 out of 4 (2/4) because we assume one is already adopted so there are only 2 labradors left out of 4 dogs left.
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So the probability of adopting two labradors is P(first lab) x P(second lab) = (â…—) x (2/4) = 20%
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