Answer:
He should play the Slow Ball Challenge
Step-by-step explanation:
The number of pitches = 9 pitches
The speed of the 9 pitches = 60 mph
The percentage of the time Diego estimates he can each pitch = 95%
P = 0.95
The amount Diego will win if he can hit all 9 pitches = $45
The amount he will loose = $10
The Fast Ball challenge
The number of pitches = 4 pitches
The speed of the 4 pitches = 90 mph
The percentage of the time Diego estimates he can each pitch = 60%
The amount Diego will win if he can hit all 4 pitches = 60 %
The amount he will loose = $20
For the Slow Ball challenge, we have;
The probability that he hits all 9 pitches and wins the $45, is given by the binomial probability distribution as follows;
P(X) = ₙCₓ · Pˣ ·(1 - P)ⁿ⁻ˣ
Therefore, we get;
P(X) = ₉C₉ · P⁹ ·(1 - P)⁹⁻⁹ = 1 × 0.95⁹ ≈ 0.63025
The probability that he losses the $45 = 1 - P(X) ≈ 1 - 0.63 = 0.36975
The expected value = 0.63025 × 45 - 0.36975 × 10 ≈ 24.66375
The expected value ≈ 24.66375
For the Fast Ball challenge, we have;
The probability that he hits all 4 pitches and wins the $80, is given by the binomial probability distribution as follows;
P(X) = ₙCₓ · Pˣ ·(1 - P)ⁿ⁻ˣ
Therefore, we get;
P(X) = ₄C₄ · P⁴ ·(1 - P)⁴⁻⁴ = 1 × 0.6⁴ ≈ 0.1296
The probability that he losses the $80 stake = 1 - P(X) ≈ 1 - 0.1296 = 0.8704
The expected value = 0.1296 × 80 - 0.8704 × 20 ≈ -7.04
The expected value ≈ -7.04.
Given that the expected value for the Fast Ball Challenge is lesser than the expected value for the Slow Ball Challenge, Diego should play the Slow Ball Challenge.
