Question

A student studying for a vocabulary test knows the meanings of 10 words from a list of 26 words. If the test contains 10 words from the study list, what is the probability that at least 8 of the words on the test are words that the student knows?

Answer 1

This can be found by using the hypergeometric distribution.
The probability that exactly 8 words are known is:
\[P(8known)=\frac{\left(\begin{matrix}10 \\ 8\end{matrix}\right)\left(\begin{matrix}16 \\ 2\end{matrix}\right)}{\left(\begin{matrix}26 \\ 10\end{matrix}\right)}\]
The probability that exactly 9 words are known is:
\[P(9known)=\frac{\left(\begin{matrix}10 \\ 9\end{matrix}\right)\left(\begin{matrix}16 \\ 1\end{matrix}\right)}{\left(\begin{matrix}26 \\ 10\end{matrix}\right)}\]
The probability that exactly 10 words are known is:
\[P(10known)=\frac{\left(\begin{matrix}10 \\ 10\end{matrix}\right)\left(\begin{matrix}16 \\ 0\end{matrix}\right)}{\left(\begin{matrix}26 \\ 10\end{matrix}\right)}\]
The probability that at least 8 words are known is the sum of the three values of probability found as above.