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 find the probabilities in this scenario, we need to consider the number of favorable outcomes (getting the desired ice cream) and the total number of possible outcomes (all the combinations of ice cream scoops).
Let's calculate the probabilities for each question:
A. Teller gets exactly what he wants.
Teller wants a dish with strawberry, banana, and vanilla. Since it doesn't matter in which order the flavors are scooped into the dish, we can consider it as one combination. Therefore, the number of favorable outcomes is 1. The total number of possible outcomes is the total number of ways to scoop 3 scoops out of 40 flavors, which can be calculated using the combination formula (nCr): C(40, 3) = 40! / (3! * (40 - 3)!) = 9,880. Therefore, the probability is 1/9,880.
B. Palmer gets exactly what he wants (order and all).
Palmer wants a cone with a specific order of chocolate mousse, cherry truffle, and double Dutch chocolate. Each scoop has only one specific flavor, so the order matters in this case. We calculate the probability by multiplying the probabilities of each scoop being the correct flavor. The probability of the first scoop being chocolate mousse is 1/40, then the second scoop being cherry truffle is 1/39, and finally, the third scoop being double Dutch chocolate is 1/38. Therefore, the probability is (1/40) * (1/39) * (1/38) ≈ 1/1,592,920.
C. Palmer does not get exactly what he wants.
The probability of not getting exactly what Palmer wants is equal to 1 minus the probability of getting exactly what he wants. Therefore, the probability is 1 - 1/1,592,920 ≈ 0.999999373.
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?
Yes, Palmer has a better chance of getting what he wants if he changes his order to a dish. In part A, we already calculated that Teller has a 1/9,880 chance of getting exactly what he wants. Palmer's chances of getting exactly what he wants in a dish will be the same as Teller's, so Palmer's chances increase from 1/1,592,920 (in part B) to 1/9,880 by changing the order to a dish.