Questions: Suppose that the New England Colonials baseball team is equally likely to win a game as not to win it. If 5 Colonials games are chosen at random, what is the probability that exactly 2 of those games are won by the Colonials? Round your response to at least three decimal places. (If necessary, consult a list of formulas.)

Solution Steps

We need to find the probability that the New England Colonials baseball team wins exactly 2 out of 5 games, given that the probability of winning a game is \( p = 0.5 \) and the probability of losing a game is \( q = 1 - p = 0.5 \).

The probability of exactly \( x \) successes (wins) in \( n \) trials (games) can be calculated using the binomial probability formula:

\[ P(X = x) = \binom{n}{x} \cdot p^x \cdot q^{n-x} \]

Where:

• \( n = 5 \) (total games)
• \( x = 2 \) (games won)
• \( p = 0.5 \) (probability of winning)
• \( q = 0.5 \) (probability of losing)

The binomial coefficient \( \binom{n}{x} \) is calculated as follows:

\[ \binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5 \times 4}{2 \times 1} = 10 \]

Now we can substitute the values into the formula:

\[ P(X = 2) = \binom{5}{2} \cdot (0.5)^2 \cdot (0.5)^{5-2} \]

Calculating this gives:

\[ P(X = 2) = 10 \cdot (0.5)^2 \cdot (0.5)^3 = 10 \cdot 0.25 \cdot 0.125 = 10 \cdot 0.03125 = 0.3125 \]

Rounding \( 0.3125 \) to three decimal places gives us \( 0.312 \).

Final Answer