A football coach is trying to decide: When a team is ahead late in the game, which strategy is better?
- Play the "regular" defense.
- Play a "prevent" defense that guards against long gains but makes short gains easier.
The coach reviews the outcomes of 100 games.
| | Win | Loss | Total |
|------------------|-----|------|-------|
| Regular defense | 41 | 9 | 50 |
| Prevent defense | 32 | 18 | 50 |
| Total | 73 | 27 | 100 |
Compare the probability of winning when playing regular defense with the probability of winning when playing prevent defense. Draw a conclusion based on your results.
A. [tex]P(\text{win }|\text{ regular })=0.33[/tex]
[tex]P(\text{win | prevent})=0.67[/tex]
Conclusion: You are more likely to win by playing regular defense.
B. [tex]P(\text{win }|\text{ regular })=0.82[/tex]
[tex]P(\text{win }|\text{ prevent})=0.64[/tex]
Conclusion: You are more likely to win by playing prevent defense.
C. [tex]P(\text{win }|\text{ regular })=0.33[/tex]
[tex]P(\text{win }|\text{ prevent})=0.67[/tex]
Conclusion: You are more likely to win by playing prevent defense.
D. [tex]P(\text{win }|\text{ regular })=0.82[/tex]
[tex]P(\text{win }|\text{ prevent})=0.64[/tex]
Conclusion: You are more likely to win by playing regular defense.
Answer (1)
Calculate the probability of winning with regular defense: P ( " w in " ∣ " re gu l a r " ) = 50 41 = 0.82 .
Calculate the probability of winning with prevent defense: P ( " w in " ∣ " p re v e n t " ) = 50 32 = 0.64 .
Compare the probabilities: 0.64"> 0.82 > 0.64 .
Conclusion: You are more likely to win by playing regular defense. D
Explanation
Analyze the problem Let's analyze the given problem. We have data from 100 football games, comparing the effectiveness of 'regular' and 'prevent' defense strategies when a team is ahead late in the game. The goal is to determine which strategy yields a higher probability of winning.
Outline the solution To determine the best strategy, we need to calculate the probability of winning with each defense type. This involves dividing the number of wins for each strategy by the total number of games played with that strategy.
Calculate probability of winning with regular defense First, let's calculate the probability of winning when using the regular defense. From the table, we see that the regular defense resulted in 41 wins out of 50 games. Therefore, the probability of winning with the regular defense is: P ( " w in " ∣ " re gu l a r " ) = 50 41 = 0.82
Calculate probability of winning with prevent defense Next, let's calculate the probability of winning when using the prevent defense. The prevent defense resulted in 32 wins out of 50 games. Therefore, the probability of winning with the prevent defense is: P ( " w in " ∣ " p re v e n t " ) = 50 32 = 0.64
Compare the probabilities and draw a conclusion Now, we compare the two probabilities. We found that P ( " w in " ∣ " re gu l a r " ) = 0.82 and P ( " w in " ∣ " p re v e n t " ) = 0.64 . Since 0.64"> 0.82 > 0.64 , the regular defense has a higher probability of winning.
State the final answer Based on our calculations, the probability of winning with the regular defense (0.82) is greater than the probability of winning with the prevent defense (0.64). Therefore, the better strategy is to use the regular defense.
Examples
In sports analytics, understanding conditional probabilities helps coaches make informed decisions. For example, knowing the likelihood of success with different defensive strategies in specific game situations allows for better real-time adjustments, potentially increasing the team's chances of winning. This type of analysis extends beyond football, applying to basketball, baseball, and other sports where strategic choices impact outcomes. By quantifying these probabilities, coaches can optimize their game plans and improve overall performance.
