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

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?

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Thu Thảo 6 months 2021-08-27T15:51:21+00:00 1 Answers 0 views 0

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    2021-08-27T15:53:06+00:00

    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.

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