Question

The Slow Ball Challenge: There will be 9 pitches, each at 60 mph. Diego estimates that he will hit each individual pitch 95% of the time. If Diego can hit all 9 pitches, he will win a total of $45; otherwise he will lose $10. The Fast Ball Challenge: There will be 4 pitches, each at 90 mph. Diego estimates that he will hit each individual pitch 60% of the time. If Diego can hit all 4 pitches, he will win a total of $80; otherwise he will lose $20. Show Proof of which challenge he should play and why?

Answer 1

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.