The probability of winning a bouncy ball exactly two times in five guesses is 0.1324 or about 13.24%.
Step-by-step explanation:
We can do this using the binomial probability formula:
P(X = k) = (n choose k) * p^k * (1 - p)^(n - k)
X is the number of successes (i.e., winning the bouncy ball), n is the number of trials (i.e., number of guesses), p is the probability of success (i.e., finding the ball under the cup), and k is the number of successes we're interested in (i.e., 2 in this case).
In this scenario, n = 5 and we need to find p.
To find p, we need to consider each guess separately. For the first guess, the probability of winning is 1/4. If the first guess is incorrect, then the ball is randomly placed under one of the cups again, and the probability of winning on the second guess is also 1/4. Similarly, for the third, fourth, and fifth guesses, the probability of winning is 1/4. So, the overall probability of winning exactly 2 times in 5 guesses is:
P(X = 2) = (5 choose 2) * (1/4)^2 * (3/4)^3
= (10) * (1/16) * (27/64)
= 0.1324
Therefore, the probability of winning a bouncy ball exactly two times in five guesses is 0.1324 or about 13.24%.