To find the probability that the New England Colonials baseball team wins exactly 4 out of 5 randomly chosen games, we can use the binomial probability formula. The binomial probability formula is used when there are fixed trials (games), each trial has only two possible outcomes (win or lose), the probability of a win is the same for each trial, and the trials are independent.
Given:
- The probability of winning a game (p) = 0.5
- The number of games chosen (n) = 5
- The number of games won (k) = 4
The binomial probability formula is:
[tex]\[ P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \][/tex]
Where:
- [tex]\(\binom{n}{k}\)[/tex] is the binomial coefficient, calculated as [tex]\(\frac{n!}{k!(n-k)!}\)[/tex].
- [tex]\( p \)[/tex] is the probability of winning a game.
- [tex]\( (1 - p) \)[/tex] is the probability of losing a game.
- [tex]\( n \)[/tex] is the total number of games.
- [tex]\( k \)[/tex] is the number of games won.
Step-by-step solution:
1. Calculate the binomial coefficient [tex]\(\binom{5}{4}\)[/tex]:
[tex]\[ \binom{5}{4} = \frac{5!}{4!(5-4)!} = \frac{5!}{4!1!} = \frac{5 \times 4!}{4! \times 1} = \frac{5}{1} = 5 \][/tex]
2. Calculate [tex]\( p^k \)[/tex]:
[tex]\[ p^k = (0.5)^4 = 0.5^4 = 0.0625 \][/tex]
3. Calculate [tex]\( (1 - p)^{n - k} \)[/tex]:
[tex]\[ (1 - p)^{n - k} = (1 - 0.5)^{5 - 4} = 0.5^1 = 0.5 \][/tex]
4. Combine these values using the binomial probability formula:
[tex]\[ P(X = 4) = \binom{5}{4} \cdot 0.5^4 \cdot 0.5 \][/tex]
[tex]\[ P(X = 4) = 5 \cdot 0.0625 \cdot 0.5 \][/tex]
5. Perform the multiplication:
[tex]\[ P(X = 4) = 5 \cdot 0.0625 \cdot 0.5 = 5 \cdot 0.03125 = 0.15625 \][/tex]
Thus, the probability that the New England Colonials will win exactly 4 out of 5 randomly chosen games is 0.156, rounded to at least three decimal places.
