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Three softball players discussed their batting averages after a game.

\begin{tabular}{|l|c|}
\hline
& Probability \\
\hline
Player 1 & [tex]$\frac{7}{11}$[/tex] \\
\hline
Player 2 & [tex]$\frac{6}{9}$[/tex] \\
\hline
Player 3 & [tex]$\frac{5}{7}$[/tex] \\
\hline
\end{tabular}

Compare the probabilities and interpret the likelihood. Which statement is true?

A. Player 1 is more likely to hit the ball than Player 2 because [tex]$P($[/tex]Player 1[tex]$)\ \textgreater \ P($[/tex]Player 2[tex]$)$[/tex]

B. Player 2 is more likely to hit the ball than Player 3 because [tex]$P($[/tex]Player 2[tex]$)\ \textgreater \ P($[/tex]Player 3[tex]$)$[/tex]

C. Player 1 is more likely to hit the ball than Player 3 because [tex]$P($[/tex]Player 1[tex]$)\ \textgreater \ P($[/tex]Player 3[tex]$)$[/tex]

D. Player 3 is more likely to hit the ball than Player 2 because [tex]$P($[/tex]Player 3[tex]$)\ \textgreater \ P($[/tex]Player 2[tex]$)$[/tex]



Answer :

GinnyAnswer
To solve this problem, we will compare the batting averages (probabilities) of the three players and determine which statements are true based on these comparisons.

First, let's interpret the batting averages provided:

1. Player 1 has a batting average of [tex]\(\frac{7}{11}\)[/tex].
2. Player 2 has a batting average of [tex]\(\frac{6}{9}\)[/tex].
3. Player 3 has a batting average of [tex]\(\frac{5}{7}\)[/tex].

Next, we look at the decimal values of these fractions to make accurate comparisons:

- Player 1's batting average: [tex]\(\frac{7}{11} \approx 0.636\)[/tex]
- Player 2's batting average: [tex]\(\frac{6}{9} \approx 0.667\)[/tex]
- Player 3's batting average: [tex]\(\frac{5}{7} \approx 0.714\)[/tex]

Now, let's compare these probabilities:

1. Compare Player 1 and Player 2:
[tex]\[ 0.636 (P(\text{Player 1})) < 0.667 (P(\text{Player 2})) \][/tex]
Therefore, the statement "Player 1 is more likely to hit the ball than Player 2" is false.

2. Compare Player 2 and Player 3:
[tex]\[ 0.667 (P(\text{Player 2})) < 0.714 (P(\text{Player 3})) \][/tex]
Therefore, the statement "Player 2 is more likely to hit the ball than Player 3" is false.

3. Compare Player 1 and Player 3:
[tex]\[ 0.636 (P(\text{Player 1})) < 0.714 (P(\text{Player 3})) \][/tex]
Therefore, the statement "Player 1 is more likely to hit the ball than Player 3" is false.

4. Compare Player 3 and Player 2:
[tex]\[ 0.714 (P(\text{Player 3})) > 0.667 (P(\text{Player 2})) \][/tex]
Therefore, the statement "Player 3 is more likely to hit the ball than Player 2" is true.

Based on these comparisons:

- The statement "Player 1 is more likely to hit the ball than Player 2 because [tex]\(P(\text{Player 1}) > P(\text{Player 2})\)[/tex]" is false.
- The statement "Player 2 is more likely to hit the ball than Player 3 because [tex]\(P(\text{Player 2}) > P(\text{Player 3})\)[/tex]" is false.
- The statement "Player 1 is more likely to hit the ball than Player 3 because [tex]\(P(\text{Player 1}) > P(\text{Player 3})\)[/tex]" is false.
- The statement "Player 3 is more likely to hit the ball than Player 2 because [tex]\(P(\text{Player 3}) > P(\text{Player 2})\)[/tex]" is true.

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