# Statistics and probability: ice cream selection

My stats test is tomorrow, and my professor emailed me a question to consider.

Ben gets a three-scoop ice cream cone. There's a .4 probability that the middle scoop is chocolate, .2 prob that the top scoop is chocolate, and .6 prob that the bottom scoop is chocolate. The different scoops are mutually independent.

What's the probability that Ben gets a cone with exactly one scoop of chocolate? (How about exactly two scoops?)

For one scoop, I initially multiplied the probs to get .048, but that seems wrong. I think I need to create a union. $(T\cap NB \cap NM) \cup (NT \cap NB \cap M) \cup (NT \cap B \cap NM).$

T=Top, B = Bottom, M = Middle, NT = Not Top, etc.

I'm not sure how to get the values, assuming this is correct. Do I need to create a Venn Diagram with three circles? How? A intersects B in 2 places, etc.

• By the way, $0.048$ is the probability of getting exactly three scoops of chocolate. – Graham Kemp Oct 19 '16 at 2:34

\begin{align} &\mathsf P((A\cap B^c\cap C^c)\cup(A^c\cap B\cap C^c)\cup(A^c\cap B^c\cap C))\\ =~& \mathsf P(A)\mathsf P(B^c)\mathsf P(C^c)+\mathsf P(A^c)\mathsf P(B)\mathsf P(C^c)+\mathsf P(A^c)\mathsf P(B^c)\mathsf P(C)\end{align}