Conditional probability
Identify restricted sample spaces, apply the conditional probability formula , and solve problems involving sequential events and independence.
Worked examples
Finding conditional probability from a restricted sample space
Straightforward
Problem
A class of 30 students sat both a maths test and a science test. 20 students passed maths, 15 passed science, and 10 passed both tests. A student who passed science is chosen at random. Find the probability they also passed maths.
1
Identify the restricted sample space.
The condition is that the student passed science. Only students who passed science are considered, so the restricted sample space has 15 outcomes.
2
Count the favourable outcomes within the restricted sample space.
Favourable outcomes are students who passed both maths and science. There are 10 such students.
3
Calculate the conditional probability.
Answer
.
Applying the conditional probability formula
Moderate
Problem
For events and : , , and .
(a) Find .
(b) State whether knowing that has occurred makes more or less likely, and explain why.
(a) Find .
(b) State whether knowing that has occurred makes more or less likely, and explain why.
1
Write the conditional probability formula.
2
Substitute the given values.
3
Calculate .
4
Compare to to answer part (b).
Since , knowing that has occurred makes more likely. The events are positively associated.
Answer
(a) . (b) Knowing occurred makes more likely, since .
Conditional probability with sequential draws
Challenging
Problem
A bag contains 4 red and 6 blue marbles. Two marbles are drawn without replacement. Let be the event 'second marble is red' and let be the event 'first marble is red'.
(a) Find .
(b) Find using the law of total probability.
(c) Determine whether and are independent.
(a) Find .
(b) Find using the law of total probability.
(c) Determine whether and are independent.
1
If the first marble is red, 9 marbles remain with 3 red:
Find and for part (a).
If the first marble is red, 9 marbles remain with 3 red:
2
Apply the conditional probability formula to find .
3
Find using the law of total probability for part (b). Consider two cases: first marble red or first marble blue.
Let = 'first marble is blue'. Then .
If the first marble is blue, 9 remain with 4 red:
Applying the law of total probability:
If the first marble is blue, 9 remain with 4 red:
Applying the law of total probability:
4
Since , the events and are not independent. Knowing the first marble was red reduces the probability the second is also red (from to ), because one red marble has already been removed.
Compare and to determine independence for part (c).
Since , the events and are not independent. Knowing the first marble was red reduces the probability the second is also red (from to ), because one red marble has already been removed.
Answer
(a) . (b) . (c) and are not independent, since .
Practise
Q1·Straightforward
In a group of 40 people, 25 prefer coffee and 20 prefer tea. A total of 10 people prefer both coffee and tea. A person who prefers tea is selected at random. What is the probability they also prefer coffee? Give your answer as a fraction in simplest form.
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Explanation
The restricted sample space contains only those who prefer tea: 20 people. Of these, 10 also prefer coffee. So
Q2·Straightforward
In a class of 25 students, 14 study French and 10 study Spanish. A total of 6 students study both French and Spanish. A student who studies Spanish is selected at random. What is the probability they also study French? Give your answer as a fraction in simplest form.
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Explanation
The restricted sample space contains all students who study Spanish: 10 students. Of these, 6 also study French. So
Q3·Straightforward
A survey of 50 people found that 30 like watching TV and 20 do not. Of those who like TV, 18 also like movies. Of those who do not like TV, 12 also like movies. A person who likes movies is selected at random. What is the probability they also like TV? Give your answer as a fraction in simplest form.
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Explanation
The total number of people who like movies is . This is the restricted sample space. Of these 30 people, 18 also like TV. So
Q4·Straightforward
Cards numbered 1 to 20 are placed in a bag and one is drawn at random. Given that the number on the card is greater than 10, what is the probability that the number is also even? Give your answer as a fraction in simplest form.
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Explanation
The restricted sample space is , which contains 10 numbers. The even numbers in this set are , giving 5 favourable outcomes. So
Q5·Moderate
For events and : and . Find .
Explanation
Using the conditional probability formula:
Q6·Moderate
For events and : and . Find .
Explanation
Rearranging the conditional probability formula:
Q7·Moderate
A bag contains 5 red, 4 blue, and 3 green balls. One ball is drawn at random. Given that the ball drawn is not green, what is the probability that it is red? Give your answer as a fraction in simplest form.
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Explanation
The balls that are not green number , forming the restricted sample space. Of these, 5 are red. So
Q8·Moderate
A survey of 60 students found that 35 exercise regularly and 25 do not. Of those who exercise regularly, 28 passed a maths exam. Of those who do not exercise regularly, 15 passed the maths exam. A student who passed the maths exam is selected at random. What is the probability they exercise regularly? Give your answer as a fraction in simplest form.
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Explanation
The total number of students who passed maths is . This is the restricted sample space. Of these 43 students, 28 exercise regularly. So
Q9·Challenging
For events and : , , and . is calculated. Which conclusion is correct?
Explanation
Using the conditional probability formula:
Since , the events are independent. Knowing that has occurred does not change the probability of .
Q10·Challenging
A bag contains 6 red and 4 blue marbles. Two marbles are drawn without replacement. Given that the first marble drawn is blue, find the probability that the second marble is red. Give your answer as a fraction in simplest form.
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Explanation
After the first blue marble is removed, 9 marbles remain in the bag: 6 red and 3 blue. The restricted sample space now has 9 equally likely outcomes. The favourable outcomes (red) number 6. So
Q11·Challenging
A standard deck of 52 cards contains 4 kings. Two cards are drawn without replacement. Given that the first card drawn is a king, find the probability that the second card is also a king. Give your answer as a fraction in simplest form.
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Explanation
After one king is removed, 51 cards remain in the deck, of which 3 are kings. The restricted sample space has 51 equally likely outcomes. So
Q12·Challenging
In a population, 10% of people have a medical condition. A screening test gives a positive result for 90% of those who have the condition and for 20% of those who do not have the condition. A person selected at random tests positive. Find the probability they actually have the condition. Give your answer as a fraction in simplest form.
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Explanation
Let = has condition and = tests positive. Given: , , .
First, find using the law of total probability:
Then apply the conditional probability formula:
Despite the positive test, there is only a probability of having the condition, because the condition is rare in the population.
First, find using the law of total probability:
Then apply the conditional probability formula:
Despite the positive test, there is only a probability of having the condition, because the condition is rare in the population.