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3 Responses
Great question! It’s super common to mix up independence with mutual exclusivity. Independence means the overlap should match the product: P(A ∩ B) = P(A)P(B). Here, P(A)P(B) = 0.6 × 0.5 = 0.3, but you’re given P(A ∩ B) = 0.25. Since 0.25 ≠ 0.3, A and B are not independent. Mutual exclusivity, on the other hand, would mean they never happen together, i.e., P(A ∩ B) = 0. So they’re definitely not mutually exclusive either, because 0.25 > 0. Another quick check: P(A|B) = 0.25/0.5 = 0.5, which is different from P(A) = 0.6-learning B happened actually makes A less likely, so there’s a mild “push apart” effect.
I like to sanity-check with a concrete picture: imagine 100 trials. Then about 60 are A, 50 are B, and 25 are both. If A and B were independent, we’d expect 0.6 × 0.5 × 100 = 30 people in the overlap, but we only see 25-so less overlap than independence predicts. You can even fill a Venn diagram: A only = 60 − 25 = 35, B only = 50 − 25 = 25, and neither = 100 − (25 + 35 + 25) = 15. Everything is consistent, and it shows A and B happen together sometimes (not mutually exclusive) but not at the rate independence would suggest.
You’re thinking about it right: independence needs P(A∩B)=P(A)P(B), and since 0.6×0.5=0.3 but you’ve got 0.25, they’re not independent (and not mutually exclusive either, since that would make P(A∩B)=0)…though 0.25 being close to 0.3 kind of makes them “almost independent,” which might just be me overthinking it.
Example: two coin flips-A: first is heads (0.5), B: second is heads (0.5), P(A∩B)=0.25=0.5×0.5; nice walkthrough here: https://www.khanacademy.org/math/statistics-probability/probability-library/independence-and-dependence/a/independent-events
Great question-it’s easy to mix up independence and mutual exclusivity. To check independence, compare P(A ∩ B) with P(A)P(B): here 0.6 × 0.5 = 0.30, but you’re given P(A ∩ B) = 0.25, so A and B are not independent. Another way to see it is via conditional probability: P(B | A) = 0.25/0.6 ≈ 0.417, which is not equal to P(B) = 0.5, so knowing A occurred changes the chance of B. For mutual exclusivity, you would need P(A ∩ B) = 0; since it’s 0.25, they’re not mutually exclusive either. Intuitively, the fact that 0.25 < 0.30 means A and B occur together less often than they would if they were independent (a kind of negative association). Hope this helps!