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If you have a 10 sided die, what is the probability that you will roll an odd number greater than 5?

I tried to approach this as P(odd)*P(greater than 5). Since the probability of rolling an odd number is (1/2) and the probability of rolling a number greater than 5 is also (1/2) the answer should be (1/4).

However, when listing the possibilities, the only odd numbers greater than 5 are 7 and 9, so this gives a probability of 2/10 = 1/5.

Why is the answer wrong when you multiply the probabilities like I first tried?

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  • $\begingroup$ This should be conditional probability. I seem to get 2/5 instead. $\endgroup$ – Hesky Cee Feb 14 '14 at 0:52
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It's because the probability that it's odd and the probability that it's greater than 5 are not independent. You can only multiply those probabilities if they're independent (source). Here, there are 3 odd numbers not greater than 5, but only 2 odd numbers greater than 5, so there's the dependence.

The probability for dependent events is $P(A) * P(B|A)$. $P(odd) = 1/2$, and $P(greaterThan5 | isOdd) = 2/5$, because out of the $5$ odd numbers, only $2$ are greater than $5$. Then $1/2 * 2/5 = 1/5$.

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