Question

The Slow Ball Challenge or The Fast Ball Challenge.

The Slow Ball Challenge: There will be 7 pitches, each at 60 mph. Diego estimates

that he will hit each individual pitch 80% of the time. If Diego can hit all 7 pitches, he

will win a total of $60; otherwise he will lose $10.

The Fast Ball Challenge: There will be 3 pitches, each at 90 mph. Diego estimates

that he will hit each individual pitch 70% of the time. If Diego can hit all 3 pitches, he

will win a total of $60; otherwise he will lose $10.

Which challenge should he choose?

Fast: 4.68

Slow: 4.68

Fast: 14.01

Slow: 14.01

G

Answer 1

Fast ball challenge

Explanation:

Given

Slow Ball Challenge

`Pitches = 7`

`P(Hit) = 80\%`

`Win = \$60`

`Lost = \$10`

Fast Ball Challenge

`Pitches = 3`

`P(Hit) = 70\%`

`Win = \$60`

`Lost = \$10`

Required

Which should he choose?

To do this, we simply calculate the expected earnings of both.

Considering the slow ball challenge

First, we calculate the binomial probability that he hits all 7 pitches

`P(x) =^nC_x * p^x * (1 - p)^(n - x)`

Where

`n = 7` --- pitches

`x = 7` --- all hits

`p = 80\% = 0.80` --- probability of hit

So, we have:

`P(x) =^nC_x * p^x * (1 - p)^(n - x)`

`P(7) =^7C_7 * 0.80^7 * (1 - 0.80)^(7 - 7)`

`P(7) =1 * 0.80^7 * (1 - 0.80)^0`

`P(7) =1 * 0.80^7 * 0.20^0`

Using a calculator:

`P(7) =0.2097152` --- This is the probability that he wins

i.e.

`P(Win) =0.2097152`

The probability that he lose is:

`P(Lose) = 1 - P(Win)` ---- Complement rule

`P(Lose) = 1 -0.2097152`

`P(Lose) = 0.7902848`

The expected value is then calculated as:

`Expected = P(Win) * Win + P(Lose) * Lose`

`Expected = 0.2097152 * \$60 + 0.7902848 * \$10`

Using a calculator, we have:

`Expected = \$20.48576`

Considering the fast ball challenge

First, we calculate the binomial probability that he hits all 3 pitches

`P(x) =^nC_x * p^x * (1 - p)^(n - x)`

Where

`n = 3` --- pitches

`x = 3` --- all hits

`p = 70\% = 0.70` --- probability of hit

So, we have:

`P(3) =^3C_3 * 0.70^3 * (1 - 0.70)^(3 - 3)`

`P(3) =1 * 0.70^3 * (1 - 0.70)^0`

`P(3) =1 * 0.70^3 * 0.30^0`

Using a calculator:

`P(3) =0.343` --- This is the probability that he wins

i.e.

`P(Win) =0.343`

The probability that he lose is:

`P(Lose) = 1 - P(Win)` ---- Complement rule

`P(Lose) = 1 - 0.343`

`P(Lose) = 0.657`

The expected value is then calculated as:

`Expected = P(Win) * Win + P(Lose) * Lose`

`Expected = 0.343 * \$60 + 0.657 * \$10`

Using a calculator, we have:

`Expected = \$27.15`

So, we have:

`Expected = \$20.48576` -- Slow ball

`Expected = \$27.15` --- Fast ball

The expected earnings of the fast ball challenge is greater than that of the slow ball. Hence, he should choose the fast ball challenge.