Practice Conditional probability
Conditional probability measures the chance of an event when another event is already known to have occurred.
Restrict the sample space to that condition.
Exercises with solutions
The conditional probability P(A|B) gives the probability of A given that B has occurred. It is calculated as P(A|B) = P(A ∩ B) ÷ P(B). Unlike P(A) it refers only to the restricted sample space B.
Exercise 1
A class has 12 girls and 8 boys; 9 girls and 5 boys play football. What is the probability that a football player is a girl?
Show solution and working
9/14
There are 9 + 5 = 14 football players, of whom 9 are girls. So 9 ÷ 14.
Exercise 2
What is the probability that a girl plays football?
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9/12 = 3/4
Of the 12 girls, 9 play football: 9 ÷ 12 = 3/4.
Exercise 3
What distinguishes P(A|B) from P(B|A)?
Show solution and working
the condition and the total it refers to
P(A|B) refers to the set B, P(B|A) to the set A. The two values are usually different.
Exercise 4
An urn holds 4 red and 6 blue balls. Two are drawn without replacement. What is the probability that the second is red given the first was red?
Show solution and working
3/9 = 1/3
After the first red ball, 9 balls remain of which 3 are red. So 3 ÷ 9 = 1/3.
Common mistake
Using the wrong total for a conditional probability. After one red ball is drawn only 9 balls remain, not 10 — exactly what distinguishes drawing with and without replacement.
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