Probability Workers D2A5Ff

1. **State the problem:** We want to find the probability that more than $40$ out of $50$ workers and/or their spouses have saved money for retirement. 2. **Identify the distribution:** This is a binomial probability problem where the number of trials $n=50$ and we want $P(X>40)$ where $X$ is the number of successes. 3. **Formula:** The binomial probability mass function is $$P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$$ where $p$ is the probability of success on a single trial. 4. **Important note:** The problem does not provide $p$, the probability that a single worker/spouse has saved money. Without $p$, we cannot calculate the exact probability. 5. **Assuming $p$ is known:** To find $P(X>40)$, we calculate $$P(X>40) = 1 - P(X \leq 40) = 1 - \sum_{k=0}^{40} \binom{50}{k} p^k (1-p)^{50-k}$$ 6. **Using normal approximation (if $p$ known and $np$, $n(1-p)$ are large):** - Mean: $$\mu = np$$ - Standard deviation: $$\sigma = \sqrt{np(1-p)}$$ - Approximate $$P(X>40) \approx P\left(Z > \frac{40.5 - \mu}{\sigma}\right)$$ where $Z$ is standard normal. 7. **Summary:** Without the value of $p$, the problem cannot be solved numerically. If $p$ is given, use the binomial formula or normal approximation to find the probability. **Final answer:** The probability depends on the unknown $p$ and cannot be computed without it.