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I'm studying for a high school test on probability and came across this problem:

Jessica bought a crate of apples from a farm. The farmer told her that when randomly grabbing an apple from the crate, there's a 0.05 probability that it is rotten.

The question: If Jessica were to randomly grab 5 apples from the crate, what is the probability that exactly 2 would be rotten?

It doesn't say how many apples are in the crate(hence my confusion) so I simply assume that there are 100 apples for example. My calculation: 95C3 * 5C2 / 100C5 which equals to about 0.018. It says here that the correct answer is 0.021. Where did I go wrong? Any help would be awesome.

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  • $\begingroup$ Perhaps they intended you to assume that the number of apples is so large that each choice is rotten (or not) independent of the other choices. (I agree that this should have been specified). $\endgroup$ – lulu Mar 24 '19 at 19:33
  • $\begingroup$ I should say: of course the answer depends on the total number of apples. If there are $20$ apples, then there is only one rotten one so the answer is $0$. As I said, I expect they want you to assume that the number is large. $\endgroup$ – lulu Mar 24 '19 at 19:35
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I see your logic, the chances of picking 2 rotten apples and 3 good apples.

I think this would work if there were exactly $100$ apples, but the probability smoothes out as you get a larger crate.

So one way you could do it is pick $3$ good apples, and then $2$ bad apples, which is what you did.

This would be accomplished with $(0.95)^3\dot(0.05)^2$ probability.

However, you have to account for all the various orders in which you could pick $3$ good apples and $2$ bad apples, so multiply by $\displaystyle \binom{5}{2}$, to get your answer of $10(0.95)^3(0.05)^2$.

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  • $\begingroup$ I think it might make more sense with coin flipping $\endgroup$ – Saketh Malyala Mar 24 '19 at 19:39
  • $\begingroup$ What are the chances of flipping a coin three times and ending with only one head? $\endgroup$ – Saketh Malyala Mar 24 '19 at 19:39
  • $\begingroup$ Well since there are three WAYS to flip a coin with one head (HTT, THT, TTH) you multiply by (3 choose 1) * (1/2) * (1/2)^2 = 3/8, which you can verify $\endgroup$ – Saketh Malyala Mar 24 '19 at 19:40
  • $\begingroup$ Thank you a lot for the correct answer and your time. I am having trouble understanding though.. Why do you multiply it by 5C2? Is it because we only need the rotten ones? $\endgroup$ – Zae Mar 24 '19 at 19:49
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    $\begingroup$ Oh never mind I got it now(I'm slow), thanks a lot again :) $\endgroup$ – Zae Mar 24 '19 at 19:53

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