Question

Team A has probability \( \frac{2}{3} \) of winning whenever it plays. Suppose A plays four games. What is the probability that A wins more than half of its games?

• A. \( \frac{16}{27} \)
• B. \( \frac{19}{81} \)
• C. \( \frac{32}{81} \)
• D. \( \frac{27}{81} \)

Hint:

Use the binomial distribution formula \( P(X = k) = \binom{n}{k} p^k (1 - p)^{n - k} \) to calculate probabilities for a fixed number of trials.

Answer 1

The Correct Option is A

Step 1: Use binomial distribution. Since the probability of winning is \( \frac{2}{3} \), we apply the binomial distribution formula to calculate the probability that A wins more than half of the games (i.e., at least 3 games).
Step 2: Conclusion. The probability that A wins more than half of the games is \( \frac{16}{27} \).