Question

Derek Jeter challenges Albert Pujols to a batting battle. Each will earn 1 point for a hit other than a home run and 4 points for a home run in each round. Which player should you back given the statistics in the table below? Note that home runs are included as hits in the table. Data for Two Batters Batter At Bats Hits Home Runs Jeter 488 156 9 Pujols 448 125 26 E(Jeter) = 0. 38 and E(Pujols) = 0. 45, so back Pujols. E(Jeter) = 0. 39 and E(Pujols) = 0. 51, so back Pujols. E(Jeter) = 0. 34 and E(Pujols) = 0. 34, so it doesn’t matter who you back. E(Jeter) = 1. 22 and E(Pujols) = 0. 94, so back Jeter.

Answer 1

The player that should be backed is specified by: Option B:  E(Jeter) = 0. 39 and E(Pujols) = 0. 51, so back Pujols.

How to find the mean (expectation) of a random variable?

Supposing that the considered random variable is discrete, we get:

$Mean = E(X) = \sum_{\forall x_i} f(x_i)x_i$

where $x_i; \: \: i = 1,2, ... ,n$ is its n data values

and $f(x_i)$ is the probability of   $X = x_i$

How to calculate the probability of an event?

Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.

Then, suppose we want to find the probability of an event E.

Then, its probability is given as

$P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}} = \dfrac{n(E)}{n(S)}$

where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.

For the considered case, the scores of Pujols and Jeter are specified as:

                Bats      Hits      Home runs

Jetter         488      156           9

Pujols        448       125           26

The points that will be earned are:

On each hit : 1 point

On each home run: 4 points

From the statistics specified , the probability of Jetter scoring hits is estimated as:

$P(\text{Jetter scoring hits})=\dfrac{\text{n(hits by Jetter)}}{\text{n(total bats for Jetter)}} = \dfrac{156}{488}$

The probability of Jetter scoring home run is estimated as:

$P(\text{Jetter scoring home runs})=\dfrac{\text{n(home runs by Jetter)}}{\text{n(total bats for Jetter)}} = \dfrac{9}{488}$

Therefore, we get:

E(Jetter's score) = Score for hits × hits probability by Jetter + Score for home runs × home run's probability by Jetter

E(Jetter's score) = $1 \times \dfrac{156}{488} + 4 \times\dfrac{9}{488} = \dfrac{192}{488} \approx 0.39$

Similarly, from the statistics specified , the probability of Pujol scoring hits is estimated as:

$P(\text{Pujols scoring hits})=\dfrac{\text{n(hits by Pujols)}}{\text{n(total bats for Pujols)}} = \dfrac{125}{448}$

The probability of Pujol scoring home run is estimated as:

$P(\text{Pujols scoring home runs})=\dfrac{\text{n(home runs by Pujols)}}{\text{n(total bats for Pujols)}} = \dfrac{26}{448}$

E(Pujol's score) = Score for hits × hit's probability by Pujol + Score for home runs × home run's probability by Pujol

E(Pujol's score) = $1 \times \dfrac{125}{448} + 4 \times\dfrac{26}{448} = \dfrac{229}{448} \approx 0.51$

Thus, the player that should be backed is specified by: Option B:  E(Jeter) = 0. 39 and E(Pujols) = 0. 51, so back Pujols.

Learn more about expectation here:

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