An ice cream vendor sells three flavors: chocolate, strawberry, and vanilla. Forty five percent of the sales are chocolate, while 30% are strawberry, with the rest vanilla flavored. Sales are by the cone or the cup. The percentages of cones sales for chocolate, strawberry, and vanilla, are 75%, 60%, and 40%, respectively. For a randomly selected sale, define the following events:

Question

= chocolate chosen
 = strawberry chosen
 = vanilla chosen
 = ice cream on a cone
ice cream in a cup

Find the probability that the ice cream was strawberry flavor, given that it was sold in a cup.

Bot · Answer

Let S be the event that the ice cream is strawberry flavored, and let C be the event that it is in a cup. We want to find the probability P(S|C), which is the probability of the ice cream being strawberry flavored given that it is sold in a cup.

We are given the probability of each flavor:

P(chocolate) = 0.45
P(strawberry) = 0.30
P(vanilla) = 0.25

We are also given the probability of cone sales for each flavor:

P(cone | chocolate) = 0.75
P(cone | strawberry) = 0.60
P(cone | vanilla) = 0.40

Since the probabilities of cone sales and cup sales must add up to 1 for each flavor, we can find the probability of cup sales for each flavor:

P(cup | chocolate) = 1 - P(cone | chocolate) = 0.25
P(cup | strawberry) = 1 - P(cone | strawberry) = 0.40
P(cup | vanilla) = 1 - P(cone | vanilla) = 0.60

Now we can apply the definition of conditional probability to find the desired probability:

P(S|C) = P(S ∩ C) / P(C)

We can find P(S ∩ C) using the formula P(S ∩ C) = P(C | S) * P(S):

P(S ∩ C) = P(cup | strawberry) * P(strawberry) = 0.40 * 0.30 = 0.12

To find P(C), we can use the law of total probability:

P(C) = P(C | chocolate) * P(chocolate) + P(C | strawberry) * P(strawberry) + P(C | vanilla) * P(vanilla) = 0.25 * 0.45 + 0.40 * 0.30 + 0.60 * 0.25 = 0.1125 + 0.12 + 0.15 = 0.3825

Now we can find the desired probability:

P(S|C) = P(S ∩ C) / P(C) = 0.12 / 0.3825 ≈ 0.3137

So the probability that the ice cream was strawberry flavor, given that it was sold in a cup, is approximately 0.3137.