Question 2 (a)
Total probability that an investment lost value
4 marks
- Time to write
- About 8 minutes
Question
Question 2 (a) · 4 marksAn investor manages his portfolio using either mutual funds or stocks. If he invests in mutual funds, the probability that his investment loses value is 0.15, and if he invests in stocks, the probability of its losing value is 0.3. What is the probability that the investment lost value?
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
What the paper leaves open
As printed, the question gives no probability for choosing between mutual funds and stocks; it is read as equally likely.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Events and the equally likely assumption 1 mark 2 min
- 2 Joint probabilities by the multiplication theorem 2 marks 3 min
- 3 Total probability 1 mark 2 min
Short on time? Step 2 carries 2 of the 4 marks: guard its minutes.
Try it first. About 8 minutes for 4 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.
Events. Let M be the event that the investor uses mutual funds, S that he uses stocks, and L that the investment loses value. He uses one or the other, so M and S are mutually exclusive and exhaustive. As the paper note records, the question gives no chance of choosing either, so they are taken as equally likely: P(M) = P(S) = 0.5. Given: P(L | M) = 0.15 and P(L | S) = 0.3.
Total probability. L can happen only together with M or with S, and the two joint events are mutually exclusive, so P(L) = P(M ∩ L) + P(S ∩ L). Each joint probability comes from the multiplication theorem, P(M ∩ L) = P(M) × P(L | M), and the addition rule for mutually exclusive events adds them.
| Route | P(route) | P(L | route) | Joint probability |
|---|---|---|---|
| Mutual funds | 0.5 | 0.15 | 0.075 |
| Stocks | 0.5 | 0.3 | 0.15 |
| Total | 0.225 |
P(L) = 0.5 × 0.15 + 0.5 × 0.3 = 0.075 + 0.15 = 0.225.
There is a 22.5% chance that the investment lost value. (If the investor's choice were weighted differently, the answer would move between 0.15 and 0.3 accordingly.)
Where the 4 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Events and the equally likely assumption | 1 |
| 2 | Joint probabilities by the multiplication theorem | 2 |
| 3 | Total probability | 1 |
| Total | 4 |
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