Question 2 OR (f)
Probability that both analysts solve the problem
6 marks
- Time to write
- About 12 minutes
Question
Question 2 OR (f) · 6 marksTwo financial analysts, A and B, are set a complicated financial problem. The probabilities of A and B solving the problem correctly are 0.6 and 0.4, respectively. What is the probability that the problem is solved correctly by both analysts?
Reworded from the university paper: names and figures differ, the task is the same. The original paper belongs to the University of Delhi. See DU's question bank.
Before you write
Read off the question and its marks
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Events and the independence assumption 2 marks 3 min
- 2 Multiplication theorem and P both 3 marks 5 min
- 3 Check through the complete set of outcomes 1 mark 2 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Short on time? Step 2 carries 3 of the 6 marks: guard its minutes.
Try it first. About 12 minutes for 6 marks. Write it out, then open the solution.
Solution
Hover or tap a marked figure, like 72,000, to see how it is made and where else it appears.
Assumption. Let A be the event that analyst A solves the problem and B that analyst B does: P(A) = 0.6, P(B) = 0.4. The analysts work on the problem separately, so whether one succeeds does not change the other's chance: A and B are taken as independent events. For independent events P(A ∩ B) = P(A) × P(B).
Probability that both solve it.
P(A ∩ B) = P(A) × P(B) = 0.6 × 0.4 = 0.24.
There is a 24% chance that the problem is solved correctly by both analysts.
The full picture. The same rule, applied to the complements (independent too), gives every outcome:
| Outcome | Working | Probability |
|---|---|---|
| Both solve it | 0.6 × 0.4 | 0.24 |
| Only A solves it | 0.6 × 0.6 | 0.36 |
| Only B solves it | 0.4 × 0.4 | 0.16 |
| Neither solves it | 0.4 × 0.6 | 0.24 |
| Total | 1 |
So the problem is solved by at least one analyst with probability 1 − 0.24 = 0.76, by exactly one with probability 0.36 + 0.16 = 0.52, and by both with probability 0.24. If the analysts consulted each other, the events would no longer be independent and the product rule would not apply.
Where the 6 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Events and the independence assumption | 2 |
| 2 | Multiplication theorem and P(both) | 3 |
| 3 | Check through the complete set of outcomes | 1 |
| Total | 6 |
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