Conditional Probability

You may well have tried the Monty Hall problem, a question based on the game show 'Let's Make a Deal'. In the game show, extra information is given about the whereabouts of a prize. If you use this extra information, you are more likely to choose the correct strategy and win the car! This is the basis of conditional probability: extra information changes probabilities. You should be able to recognise these sort of questions, as they often contain the words "given that".


Key Concepts

On this page, you should learn about

  • conditional probability
  • using Venn diagrams to visualise conditional probability, the probability of A given B

\(P(A|B)=\frac { P(A\cap B) }{ P(B) } \)

\(P(A|B)=\frac { c }{ c+b } \)

  • Bayes theorem

Summary

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Test Yourself

You can practise some of the skills in this page with the following quiz


START QUIZ!

Exam-style Questions

Question 1

Henri Tarr travels to school by bike \(\frac{3}{5}\) of the week and by car the rest of the time.

If he travels by bike, the probability that he is late is \(\frac{1}{8}\).

If he travels by car, the probability that he is late is \(\frac{3}{8}\).

a) Copy and complete the following tree diagram

b) Find the probability that Henri goes to school by car and is late for school.

c) Find the probability that he is late for school.

d) Given that he is late, find the probability that he travels to school by car.


Hint

Full Solution

Question 2

A and B are independent events. P(A) = 0.3 and P(B) = 0.4

a) Find \(P(A'\cap B')\)

b) Hence find P(A'|B')


Hint

Full Solution

Question 3

In the Olympic Games, it is believed that 5% of sprinters are taking anabolic steroids. A test is developed that gives a positive result for an athlete who has taken anabolic steroids in the last month in 90% of cases. The test gives a positive result for an athlete who has not taken anabolic steroids in the last month in 15% of cases. A sprinter in chosen at random.

a) Find the probability that she/he tests positive anabolic steroids

b) Given that the athlete tests positive, find the probability that the athlete has taken anabolic steroids in the last month.


Hint

Full Solution

Question 4

A and B are independent events.

P(A) = P(B) = p

Show that P(A'|B')=1-p


Hint

Full Solution

MY PROGRESS

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