Answer:
(a) To find the probability that Miguel's score on the Water Hole is at most 5, we need to sum the probabilities of scoring 3, 4, and 5.
P(X ≤ 5) = P(X = 3) + P(X = 4) + P(X = 5)
= 0.15 + 0.40 + 0.25
= 0.80
Therefore, the probability that Miguel's score on the Water Hole is at most 5 is 0.80.
(b) To calculate the expected value of X, we multiply each score by its corresponding probability and sum the products.
E(X) = (3 * 0.15) + (4 * 0.40) + (5 * 0.25) + (6 * 0.15) + (7 * 0.05)
= 0.45 + 1.60 + 1.25 + 0.90 + 0.35
= 4.55
The expected value of X is 4.55. This means that, on average, Miguel can expect to score around 4.55 on the Water Hole.
(c) Let's compare the expected values of the short hit and the long hit. If the long hit is successful with a probability of 0.4, the expected value becomes 4.2. If the long hit fails and the ball lands in the water (probability of 0.6), the expected value becomes 5.4.
Expected value of short hit = 4.55
Expected value of long hit = (0.4 * 4.2) + (0.6 * 5.4)
= 1.68 + 3.24
= 4.92
Comparing the expected values, we see that the long hit with a success probability of 0.4 is better in terms of improving the expected value of the score. The expected value of 4.92 for the long hit is higher than the expected value of 4.55 for the short hit.
(d) To determine the values of p that make the long hit better than the short hit in terms of improving the expected value of the score, we need to compare the expected values.
Expected value of short hit = 4.55
Expected value of long hit = (p * 4.2) + ((1 - p) * 5.4)
To make the long hit better, we need the expected value of the long hit to be greater than 4.55.
(p * 4.2) + ((1 - p) * 5.4) > 4.55
Simplifying the inequality:
4.2p + 5.4 - 5.4p > 4.55
-1.2p > 4.55 - 5.4
-1.2p > -0.85
p < (-0.85) / (-1.2)
p < 0.7083...
Therefore, for values of p less than approximately 0.7083, the long hit will be better than the short hit in terms of improving the expected value of the score.
Step-by-step explanation: