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AQA GCSE · Question 14.2 · Probability
40% of people who fail at least one section take the test again. 5000 people take the test. How many of these 5000 people are expected to take the test again?
40% of people who fail at least one section take the test again. 5000 people take the test. How many of these 5000 people are expected to take the test again?
How to approach this question
1. First, calculate the probability that a person fails at least one section. You can do this by finding the probability they pass BOTH sections and subtracting from 1.
2. To find P(Pass A and Pass B), multiply the probabilities along the top branches of your tree diagram.
3. Once you have P(fail at least one), multiply this by the total number of people (5000) to find how many people are expected to fail at least one section.
4. Finally, calculate 40% of this number to find how many are expected to take the test again.
Full Answer
1. **Find the probability of failing at least one section.**
This is the same as 1 minus the probability of passing both sections.
P(Pass A and Pass B) = P(Pass A) × P(Pass B | Pass A) = 0.85 × 0.78 = 0.663.
P(Fail at least one) = 1 - P(Pass A and Pass B) = 1 - 0.663 = 0.337.
*Alternatively, add the probabilities of the other three outcomes:*
P(Pass A, Fail B) = 0.85 × 0.22 = 0.187
P(Fail A, Pass B) = 0.15 × 0.64 = 0.096
P(Fail A, Fail B) = 0.15 × 0.36 = 0.054
P(Fail at least one) = 0.187 + 0.096 + 0.054 = 0.337.
2. **Find the number of people who fail at least one section.**
Number = 5000 × P(Fail at least one) = 5000 × 0.337 = 1685 people.
3. **Find the number of people who take the test again.**
This is 40% of the people who failed at least one section.
Number = 1685 × 40% = 1685 × 0.4 = 674 people.
**Final answer: 674**
Step 1: Calculate the probability of failing at least one section.
The opposite of "failing at least one section" is "passing both sections". It's often easier to calculate the probability of this single outcome and subtract from 1.
The path for passing both is Pass A then Pass B. To find the probability of this combined event, we multiply the probabilities along the branches:
P(Pass A and Pass B) = P(Pass A) × P(Pass B | Pass A) = 0.85 × 0.78 = 0.663.
Now, the probability of failing at least one section is:
P(fail at least one) = 1 - P(Pass A and Pass B) = 1 - 0.663 = 0.337.
Step 2: Calculate the number of people expected to fail at least one section.
Total people = 5000.
Number failing at least one = 5000 × 0.337 = 1685.
Step 3: Calculate how many of these take the test again.
40% of those who fail at least one section take it again.
Number taking test again = 40% of 1685 = 0.40 × 1685 = 674.
So, 674 people are expected to take the test again.
Common mistakes
✗ Only calculating the probability of failing both sections, instead of failing at least one.
✗ Calculating 40% of the total 5000 people.
✗ Adding probabilities instead of multiplying along the branches (e.g., 0.85 + 0.78).
✗ Making calculation errors.
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