Falcons Relay C6725E

1. **Problem Statement:** The Falcons swim team wins 80% of their relay races and will compete in 6 races this season. We define $W$ as the number of races won. 2. **Possible values of $W$:** Since $W$ counts the number of wins out of 6 races, $W$ can be any integer from 0 to 6 inclusive. $$W \in \{0,1,2,3,4,5,6\}$$ 3. **Probability model:** This is a binomial distribution problem where the number of trials $n=6$ and probability of success (win) $p=0.8$. The probability mass function (PMF) for binomial distribution is: $$P(W=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $k$ is the number of wins. 4. **Calculate probability of exactly 5 wins:** $$P(W=5) = \binom{6}{5} (0.8)^5 (0.2)^1$$ Calculate the binomial coefficient: $$\binom{6}{5} = \frac{6!}{5!1!} = 6$$ So, $$P(W=5) = 6 \times (0.8)^5 \times 0.2$$ Calculate powers: $$(0.8)^5 = 0.32768$$ Multiply: $$P(W=5) = 6 \times 0.32768 \times 0.2 = 6 \times 0.065536 = 0.393216$$ 5. **Calculate probability that the team will lose 1 or fewer races:** Losing 1 or fewer races means winning at least 5 races (since total races = 6). So, $$P(W \geq 5) = P(W=5) + P(W=6)$$ Calculate $P(W=6)$: $$P(W=6) = \binom{6}{6} (0.8)^6 (0.2)^0 = 1 \times (0.8)^6 \times 1 = (0.8)^6$$ Calculate: $$(0.8)^6 = 0.262144$$ Sum probabilities: $$P(W \geq 5) = 0.393216 + 0.262144 = 0.65536$$ 6. **Calculate the mean number of races won:** The mean (expected value) for a binomial distribution is: $$\mu = n p = 6 \times 0.8 = 4.8$$ **Final answers:** - a) Possible values of $W$ are $\{0,1,2,3,4,5,6\}$. - b) $P(W=5) = 0.393216$ - c) $P(W \geq 5) = 0.65536$ - d) Mean number of wins $= 4.8$