Question

Of 25 people invited to a birthday party, 5 prefer vanilla ice cream, 8 prefer chocolate, and 4
prefer strawberry. The host surveys 6 of these people at random to determine how much ice
cream to buy.
a. What is the probability that at least 3 of the people surveyed prefer chocolate ice
cream?
b. What is the probability that none prefer vanilla?
c. What is the expected number of people who prefer strawberry?
d. What is the expected number of people who do not have preference for any of the three
flavors?

Answer 1

at least three means 3, 4, 5, or 6 kind of annoying, but we can do it

Answer 2

none prefer vanilla looks easiest, lets try that first
the number of ways to pick 6 people out of 25 to survey is \(\dbinom{25}{6}=177100\) an that will be the denominator

Answer 3

20 of the 25 do not prefer vanilla, and the number of ways you can pick 6 out of those 20 is \(\dbinom{20}{6}=37460\) so the answer to B is
\[\frac{\dbinom{20}{6}}{\dbinom{25}{6}}=\frac{1938}{8855}\]