Final answer:
To find the probabilities in the leaf transmutation experiment, we use combinations. (a) The probability that exactly one leaf has undergone both transformations is 3/16. (b) The probability that at least one leaf has undergone both transformations is 37/64. (c) The probability that exactly one leaf has undergone one but not both transformations is 3/8.
Step-by-step explanation:
To determine the probabilities in this leaf transmutation experiment, we need to use the concept of combinations. Since there are four possible outcomes for each leaf, the total number of outcomes is 4^3 = 64. We will calculate the probabilities for each situation:
(a) To find the probability that exactly one leaf has undergone both types of transformations, we need to choose one leaf out of three and that leaf should have undergone both types of transformations. The probability can be calculated as: (3 choose 1) * (1/4)^1 * (3/4)^2 = 3/16
= 0.1875.
(b) To find the probability that at least one leaf has undergone both transformations, we can calculate the complement probability of none of the leaves undergoing both types of transformations. The probability can be calculated as: 1 - P(No leaf has undergone both transformations) = 1 - (3/4)^3
= 37/64
= 0.5781.
(c) To find the probability that exactly one leaf has undergone one but not both transformations, we need to choose one leaf out of three and it should have undergone exactly one type of transformation. The probability can be calculated as: (3 choose 1) * (1/2)^1 * (1/2)^2 = 3/8
= 0.375.