Question 2 (a)
Two projects: probability that Chennai also misses, and that at least one succeeds
5 marks
- Time to write
- About 10 minutes
Question
Question 2 (a) · 5 marksAt present a company is running two projects, one in Pune and the other in Chennai. The probability that the Pune is successful is 0.3, while the Chennai project is successful with probability 0.6.
If the Pune project is not successful, what is the probability that the Chennai project is not successful either?
What is the probability of at least one of the two projects being successful?
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
Read from the question
- Is…?What is the…?Two asks in one question: answer both.
What the paper leaves open
As printed, the first probability sentence names the first city without the word "project" ("the [city] is successful"); it is read as that city's project. The paper gives no joint probability and does not say whether one result affects the other; the two projects, run in different cities, are taken as independent.
Write it in this order
- Read the data; note exactly what is asked 1 min
- 1 Events, complements and the independence assumption 1 mark 2 min
- 2 (i) Conditional probability that Chennai is also unsuccessful 2 marks 3 min
- 3 (ii) Probability that at least one succeeds, with the check 2 marks 3 min
- Check: arithmetic, units (₹, %), final answer underlined 1 min
Try it first. About 10 minutes for 5 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 and assumption. Let A be the event that the Pune project succeeds and B the event that the Chennai project succeeds: P(A) = 0.3 and P(B) = 0.6, so P(A′) = 1 − 0.3 = 0.7 and P(B′) = 1 − 0.6 = 0.4, since an event and its complement together have probability 1. The two projects run in different cities and the question gives no figure linking them, so their outcomes are taken as independent: knowing how one project went does not change the chance of the other, and P(A ∩ B) = P(A) × P(B). The complements of independent events are independent too.
(i) Chennai unsuccessful, given that Pune is unsuccessful
By the conditional probability rule, P(B′ | A′) = P(A′ ∩ B′) / P(A′):
P(B′ | A′) = (0.7 × 0.4) / 0.7 = 0.28 / 0.7 = 0.4.
Because the projects are independent, the Pune result carries no information about Chennai: the chance that Chennai is also unsuccessful stays at its own value, 0.4.
(ii) At least one project successful
At least one succeeds is the complement of neither succeeds:
P(A ∪ B) = 1 − P(A′ ∩ B′) = 1 − 0.7 × 0.4 = 1 − 0.28 = 0.72.
Check by the addition theorem for any two events: P(A ∪ B) = P(A) + P(B) − P(A ∩ B) = 0.3 + 0.6 − 0.18 = 0.72.
| Outcome | Working | Probability |
|---|---|---|
| Both succeed | 0.3 × 0.6 | 0.18 |
| Only Pune succeeds | 0.3 × 0.4 | 0.12 |
| Only Chennai succeeds | 0.7 × 0.6 | 0.42 |
| Neither succeeds | 0.7 × 0.4 | 0.28 |
| Total | 1 |
At least one succeeds in the first three rows: 0.18 + 0.12 + 0.42 = 0.72, as in (ii).
The other reading. If (i) were read as asking for the chance that both projects are unsuccessful (a joint, not a conditional, probability), the answer would be P(A′ ∩ B′) = 0.7 × 0.4 = 0.28. The wording "if the Pune project is not successful" states a condition, so the conditional figure 0.4 is the answer given above.
Where the 5 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Events, complements and the independence assumption | 1 |
| 2 | (i) Conditional probability that Chennai is also unsuccessful | 2 |
| 3 | (ii) Probability that at least one succeeds, with the check | 2 |
| Total | 5 |
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