2.7. Conditional Probability http://www.ck12.org
When working with events that are conditionally probable, you are working with 2 events, where the probability of
the second event is conditional on the first event occurring. Say, for example, that you want to know the probability
of drawing 2 kings from a deck of cards. As we have previously learned, here is how you would calculate this:
P(first king) =1
13
P(second king) =3
51
P(2 kings) =1
13
×
3
51
P(2 kings) =3
663
P(2 kings) =1
221
Now let’s assume you are playing a game where you need to draw 2 kings to win. You draw the first card and get a
king. What is the probability of getting a king on the second card? The probability of getting a king on the second
card can be thought of as a conditional probability. The formula for calculating conditional probability is given as:
P(B|A) =
P(A∩B)
P(A)
P(A∩B) =P(A)×P(B|A)
Another way to look at the conditional probability formula is as follows. Assuming the first event has occurred, the
probability of the second event occurring is:
P(second event|first event) =
P(first event and second event)
P(first event)Let’s work through a few problems using the formula for conditional probability.
Example A
You are playing a game of cards where the winner is determined when a player gets 2 cards of the same suit. You
draw a card and get a club(♣). What is the probability that the second card will be a club?
Step 1:List what you know.
First event = drawing the first club
Second event = drawing the second club