Question

A student studying for a vocabulary test knows the meanings of 16 words from a list of 20 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

at least 8 means 8 or 9 or 10, so you have a bit of computing to do

Answer 2

Is ti just 8/10 * 9/10 * 10/10?

Answer 3

lets compute the probability that you get exactly 8 words you know
total number of ways to pick 10 words out of 20 is \(\dbinom{20}{10}= 184756\)

Answer 4

no, that will not work it is a bit harder

Answer 5

now you want exactly 8 out of the 16 you know, and 2 out of the 4 you do not, number of ways to pick 8 out of 16 is \(\dbinom{16}{8}= 12870\) and the number of ways to choose 2 out of 4 is \(\dbinom{4}{2}=12\) so answer to the probability you get exactly 8 you know is
\[\frac{\dbinom{16}{8}\dbinom{4}{2}}{\dbinom{20}{10}}\]
\[=\frac{12870\times 12}{184756}=\frac{270}{323}\]

Answer 6

That makes sense

Answer 7

similarly exactly 9 you compute via
\[\frac{\dbinom{16}{9}\times \dbinom{4}{1}}{\dbinom{20}{10}}\]

Answer 8

clear right? numerator is 9 out of the 16 you know, 1 out of the 4 you do not, and denominator is the number of ways to choose 10 out of 20

Answer 9

Very clear

Answer 10

last one is
\[\frac{\dbinom{16}{10}}{\dbinom{20}{10}}\] and then you have to add all this crap up
have fun!

Answer 11

Haha thanks

Answer 12

yw.
i do not think there is another snappier way to do this, but i could be wrong