Palmer and teller go to the local ice cream parlor after work that offers 40 flavors of ice cream. Palmer wants a cone with chocolate mousse on top, cherry truffle in the middle, and double Dutch chocolate on the bottom. Teller wants a dish with strawberry, banana, and vanilla. Just when they finish ordering, the lights go off and it is completely dark. The ice cream scooper can find the dishes and the cones, but must scoop the ice cream at random in the dark. What is the probability that:
A. Teller gets exactly what he wants?
B. Palmer gets exactly what he wants (order and all)?
C. Palmer does not get exactly what he wants?
D. If Palmer changes his order keeping the same ice cream flavors, but asking for them in a dish, does he have a better chance of getting what he wants? Explain.
To answer these questions, we need to calculate the probabilities based on the given conditions. First, let's determine the total number of possible outcomes, which is the number of ways the ice cream scooper could randomly scoop the ice cream.
A. To calculate the probability that Teller gets exactly what he wants, we need to find the number of favorable outcomes (where Teller's order is scooped in his dish) divided by the total number of possible outcomes.
Teller's order consists of three flavors: strawberry, banana, and vanilla. There are 40 flavors in total, so the probability of Teller getting his first flavor (strawberry) is 1/40. Since the scooper must randomly find the flavors in the dark, the probability of getting his second flavor (banana) is also 1/40. Finally, the probability of getting his third flavor (vanilla) is 1/40. To calculate the probability of Teller getting exactly what he wants, we multiply these probabilities together:
P(Teller gets exactly what he wants) = (1/40) * (1/40) * (1/40)
B. To calculate the probability that Palmer gets exactly what he wants, we use a similar process as above. Palmer's order consists of three specific flavors: chocolate mousse, cherry truffle, and double Dutch chocolate. Again, assuming the scooper randomly finds each flavor with equal probability, we get:
P(Palmer gets exactly what he wants) = (1/40) * (1/40) * (1/40)
C. To find the probability that Palmer does not get exactly what he wants, we can subtract the probability of him getting what he wants from 1. This accounts for all the other possible outcomes where his order is not precisely scooped.
P(Palmer does not get exactly what he wants) = 1 - P(Palmer gets exactly what he wants)
D. If Palmer changes his order to a dish, the probability of getting what he wants may change. To calculate this, we need to consider how the order of the ice cream scoops affects the outcome.
In a dish, Palmer's order becomes strawberry, banana, and vanilla. The scooper can randomly choose any of the three flavors first, then any of the remaining two flavors second, and the last flavor third. This results in 3! (3 factorial) different ways to scoop the flavors.
Therefore, the probability of Palmer getting exactly what he wants in a dish is:
P(Palmer gets exactly what he wants in a dish) = 1 / 3!
Comparing this probability to the probability of Palmer getting exactly what he wants in a cone (as calculated in part B), you can determine if he has a better chance in a dish.
Please note that without knowing the specific rules or constraints of the ice cream parlor (such as flavors availability, cone or dish limitations, or any other relevant factors), we are making assumptions to calculate the probabilities based on the information provided.