Questions: The Ludlow Wildcats baseball team, a minor league team in the Cleveland Indians organization, plays 70% of their games at night and 30% during the day. The team wins 50% of their night games and 90% of their day games. According to today's newspaper, they won yesterday. What is the probability the game was played at night? Note: Round your answer to 4 decimal places.

Solution Steps

To find the probability that the game was played at night given that they won, we can use Bayes' Theorem. First, calculate the probability of winning a game overall. Then, use Bayes' Theorem to find the probability that the game was played at night given that they won.

Let:

• \( P(\text{Night}) = 0.70 \)
• \( P(\text{Day}) = 0.30 \)
• \( P(\text{Win} | \text{Night}) = 0.50 \)
• \( P(\text{Win} | \text{Day}) = 0.90 \)

The total probability of winning, \( P(\text{Win}) \), can be calculated using the law of total probability: \[ P(\text{Win}) = P(\text{Night}) \cdot P(\text{Win} | \text{Night}) + P(\text{Day}) \cdot P(\text{Win} | \text{Day}) \] Substituting the values: \[ P(\text{Win}) = (0.70 \cdot 0.50) + (0.30 \cdot 0.90) = 0.35 + 0.27 = 0.62 \]

To find the probability that the game was played at night given that they won, we use Bayes' Theorem: \[ P(\text{Night} | \text{Win}) = \frac{P(\text{Win} | \text{Night}) \cdot P(\text{Night})}{P(\text{Win})} \] Substituting the known values: \[ P(\text{Night} | \text{Win}) = \frac{0.50 \cdot 0.70}{0.62} = \frac{0.35}{0.62} \approx 0.564516129032258 \]

Rounding the result to four decimal places gives: \[ P(\text{Night} | \text{Win}) \approx 0.5645 \]

Final Answer