# Basic Math Question for Test

My Professor gave us twenty terms to study for an exam.

He said there will be only 10 of the 20 terms on the actual exam.

How many terms do I need to remember?

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"How many terms do I need to remember?" - how many terms do you want to be able to answer? If 3/6 is a pass and you're just looking to pass, the answer will be different than if you want to know all 6. –  crf Dec 15 '13 at 0:22
what's the passing grade –  ashley Dec 15 '13 at 0:24
@crf - funny, your comment after mine time-wise appears before mine. –  ashley Dec 15 '13 at 0:30
@ashley Actually, I think it's a glitch. Now it appears that his was before yours. –  Shahar Dec 15 '13 at 0:39
@Shahar he added the part out of the double quotations-- the rest of his first phrase after mine and this is annoying if nothing else. –  ashley Dec 15 '13 at 0:43

If you remember $16$ terms then it is certain that you will have remembered at least $6$ of the $10$ terms which are on the exam.

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To guarantee k out of the total 10 terms, remember 10+k of them and leave out the 10-k.

So, if you're looking to guarantee 4 out of 6 terms, you're looking to remember 14 terms.

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There are $\binom{20}{10}=184756$ possible exams you may receive. Of the 20 terms, suppose that you learn $K$ of them. Denote by $X$ the number of terms out of the $K$ that you know that the teacher ends up picking. Then $$P(X=x)=\frac{\binom{K}{x}\binom{20-K}{10-x}}{\binom{20}{10}}$$

I whipped up a quick table in excel

| Questions you learn/P(X≥x) | 0     | 1     | 2     | 3     | 4     | 5     | 6     |
|----------------------------|-------|-------|-------|-------|-------|-------|-------|
| 0                          | 1.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 1                          | 1.000 | 0.500 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |
| 2                          | 1.000 | 0.763 | 0.237 | 0.000 | 0.000 | 0.000 | 0.000 |
| 3                          | 1.000 | 0.895 | 0.500 | 0.105 | 0.000 | 0.000 | 0.000 |
| 4                          | 1.000 | 0.957 | 0.709 | 0.291 | 0.043 | 0.000 | 0.000 |
| 5                          | 1.000 | 0.984 | 0.848 | 0.500 | 0.152 | 0.016 | 0.000 |
| 6                          | 1.000 | 0.995 | 0.930 | 0.686 | 0.314 | 0.070 | 0.005 |
| 7                          | 1.000 | 0.998 | 0.971 | 0.825 | 0.500 | 0.175 | 0.029 |
| 8                          | 1.000 | 1.000 | 0.990 | 0.915 | 0.675 | 0.325 | 0.085 |
| 9                          | 1.000 | 1.000 | 0.997 | 0.965 | 0.815 | 0.500 | 0.185 |
| 10                         | 1.000 | 1.000 | 0.999 | 0.988 | 0.911 | 0.672 | 0.328 |
| 11                         | 1.000 | 1.000 | 1.000 | 0.997 | 0.965 | 0.815 | 0.500 |
| 12                         | 1.000 | 1.000 | 1.000 | 1.000 | 0.990 | 0.915 | 0.675 |
| 13                         | 1.000 | 1.000 | 1.000 | 1.000 | 0.998 | 0.971 | 0.825 |
| 14                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.995 | 0.930 |
| 15                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 0.984 |
| 16                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| 17                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| 18                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| 19                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
| 20                         | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |

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