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My teacher is providing us with 7 topics to study from for an upcoming midterm. The teacher will pick 4 of the 7 topics at random to be on the test. If I study 4 of the 7 topics, I know I will have a 100% chance that at least one of the topics I study will be on the upcoming midterm.

What I don't understand how to calculate are the following:

1) What are the odds that 2 out of the 4 topics I studied will appear on the midterm?

2) What are the odds, if I studied only 3 out of 7 topics, that at least 1 of the 3 topics I studied will be on the midterm exam?

For #1, above, I've tried the following: (1/7) * (1/7) * (4/7) = 0.0117 (this intuitively seems like too low of a probability).

For #2, above, is it not (3/7)*(4/7) = ~24.5% ?

If my attempts are wrong, what are the formulas/probabilities for the 2 questions above?

I greatly appreciate any help in guidance in helping me solve these problems.

Thanks!

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    @aduh Please see what I've tried in the question, now - thanks! – 2017-02-14

1 Answers 1

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Without knowing what you tried I don't want to give you the answer, but to give you some help, I think a combinatorics approach would work well here. The number of different ways of choosing 4 topics out of 7 is ${7 \choose 4}$.

For question 1 then, you'll want to think about the number of ways of picking 2 subjects from the 4 you studied and 2 subjects from the 7 - 4 = 3 that you didn't study.

For question 2, we're looking at a question that asks about at least getting 1 out of 3. Think about this in the way that getting at least 1 out of 3 is the same as 100% - getting 0 out of 3.

Hope this helps.