To find the probability that the Islanders will win at least five out of six games in a series against the Rangers, we can use the binomial probability formula.
The probability of winning a single game is \( \frac{2}{5} \), and we want to find the probability of winning five or six games out of six.
Let's calculate the probability of winning exactly five games and exactly six games, then add these probabilities together.
Probability of winning exactly five games:
\[ \text{Probability of winning 5 games} = \binom{6}{5} \times \left(\frac{2}{5}\right)^5 \times \left(\frac{3}{5}\right)^1 \]
Probability of winning exactly six games:
\[ \text{Probability of winning 6 games} = \binom{6}{6} \times \left(\frac{2}{5}\right)^6 \times \left(\frac{3}{5}\right)^0 \]
The binomial coefficient \( \binom{n}{k} \) represents the number of ways to choose k successes out of n trials.
Let's calculate these probabilities:
Probability of winning exactly five games:
\[ \binom{6}{5} = 6 \]
\[ \left(\frac{2}{5}\right)^5 \times \left(\frac{3}{5}\right)^1 = \left(\frac{32}{3125}\right) \times \left(\frac{3}{5}\right) \approx 0.0384 \]
Probability of winning exactly six games:
\[ \binom{6}{6} = 1 \]
\[ \left(\frac{2}{5}\right)^6 \times \left(\frac{3}{5}\right)^0 = \left(\frac{64}{15625}\right) \times 1 \approx 0.0041 \]
Now, add these probabilities together to find the probability of winning at least five out of six games:
\[ \text{Probability of winning at least 5 games} \approx 0.0384 + 0.0041 \approx 0.0425 \]
Rounding to the nearest thousandth, the probability that the Islanders will win at least five out of six games in a series against the Rangers is approximately 0.043.
