Question 2 (c)
Mutually exclusive events and independent events
4 marks
- Time to write
- About 8 minutes
Question
Question 2 (c) · 4 marksDistinguish between mutually exclusive events and independent events, with the help of examples.
Reworded from the university paper; 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
- examplesExamples are asked for: one short example with each point.
Write it in this order
- Read; underline the asks; jot your points 1 min
- 1 Mutually exclusive events, with example 1½ marks 3 min
- 2 Independent events, with example 1½ marks 2 min
- 3 Points of distinction 1 mark 2 min
Short on time? Trim steps 1 and 2 evenly rather than skip one: each is capped at its own marks. End with step 3 in a line or two.
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.
Mutually exclusive events. Two or more events are mutually exclusive when no two of them can happen at the same time on one trial: the happening of one rules out the others. For two such events A ∩ B is empty, P(A ∩ B) = 0, and the addition rule takes its simple form P(A ∪ B) = P(A) + P(B).
Example. One customer walks into a showroom and buys either a petrol scooter, an electric scooter, or nothing. On a single visit buys a petrol scooter and buys nothing cannot both happen.
Independent events. Two events are independent when the happening of one does not change the probability of the other: P(B | A) = P(B) (when P(A) > 0) and P(A | B) = P(A) (when P(B) > 0), so that P(A ∩ B) = P(A) × P(B); this product rule is the test of independence, and it holds even when a probability is zero.
Example. A machine in Pune breaks down and it rains in Chennai on the same day; or a card drawn from a pack is a king, and a die rolled afterwards shows a six. Knowing one result tells us nothing about the other. Independence does not need separate trials: on a single draw of one card, it is a king and it is a heart are independent, since P(king ∩ heart) = 1/52 = (4/52) × (13/52).
| Basis | Mutually exclusive events | Independent events |
|---|---|---|
| Meaning | Cannot occur together | Occurrence of one does not affect the chance of the other |
| Joint probability | P(A ∩ B) = 0 | P(A ∩ B) = P(A) × P(B) |
| Rule used | Special addition rule: P(A ∪ B) = P(A) + P(B) | Special multiplication rule |
| What it describes | Outcomes that cannot happen on the same trial | Events on the same trial or on different trials, when one carries no information about the other |
| Knowing that A has occurred (P(A) > 0) | P(B) becomes 0 | P(B) is unchanged |
The two ideas together. If A and B each have a positive probability, they cannot be both mutually exclusive and independent: exclusivity makes P(A ∩ B) = 0, while independence needs P(A ∩ B) = P(A) × P(B) > 0. In the scooter example, learning that the customer bought nothing makes the chance that they bought a scooter zero, so the events are highly dependent.
Where the 4 marks sit
| # | Component | Marks |
|---|---|---|
| 1 | Mutually exclusive events, with example | 1.5 |
| 2 | Independent events, with example | 1.5 |
| 3 | Points of distinction | 1 |
| Total | 4 |
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