Swimming Pool Services A survey found that 4 in 10 private swimming pool owners employ a pool service to care for their swimming pools. Find each probability for a sample of 10 pool owners. Round intermediate calculations and final answers to three decimal places. Part: \( 0 / 3 \) Part 1 of 3 (a) Fewer than 4 homeowners employ a pool service. \( P( \) fewer than. 4 homeowners employ a pool service) \( = \)

We are given that the probability a homeowner employs a pool service is \( p = 0.4 \) and the sample size is \( n = 10 \). Let \( X \) denote the number of homeowners (out of 10) that employ a pool service. Then, \( X \) follows a binomial distribution: \[ X \sim \text{Binomial}(n=10, p=0.4) \] We need to find the probability that fewer than 4 homeowners employ a pool service. This is \[ P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3). \] The binomial probability mass function is given by \[ P(X = k) = \binom{10}{k} \, (0.4)^k \, (0.6)^{10-k}, \] where \( \binom{10}{k} \) is the binomial coefficient. **Step 1. Compute \( P(X = 0) \):** \[ P(X = 0) = \binom{10}{0} (0.4)^0 (0.6)^{10} = 1 \times 1 \times (0.6)^{10}. \] Calculating \( (0.6)^{10} \): \[ (0.6)^{10} \approx 0.00605. \] So, \[ P(X = 0) \approx 0.006. \] **Step 2. Compute \( P(X = 1) \):** \[ P(X = 1) = \binom{10}{1} (0.4)^1 (0.6)^{9} = 10 \times 0.4 \times (0.6)^{9}. \] Calculating \( (0.6)^{9} \): \[ (0.6)^{9} \approx 0.01008. \] Thus, \[ P(X = 1) \approx 10 \times 0.4 \times 0.01008 \approx 0.0403. \] **Step 3. Compute \( P(X = 2) \):** \[ P(X = 2) = \binom{10}{2} (0.4)^2 (0.6)^{8} = 45 \times (0.16) \times (0.6)^{8}. \] Calculating \( (0.6)^{8} \): \[ (0.6)^{8} \approx 0.01680. \] Thus, \[ P(X = 2) \approx 45 \times 0.16 \times 0.01680 \approx 0.1209. \] **Step 4. Compute \( P(X = 3) \):** \[ P(X = 3) = \binom{10}{3} (0.4)^3 (0.6)^{7} = 120 \times (0.064) \times (0.6)^{7}. \] Calculating \( (0.6)^{7} \): \[ (0.6)^{7} \approx 0.02800. \] Thus, \[ P(X = 3) \approx 120 \times 0.064 \times 0.02800 \approx 0.2150. \] **Step 5. Sum the Probabilities:** \[ P(X < 4) \approx 0.00605 + 0.04031 + 0.12093 + 0.21499 \approx 0.38228. \] Rounded to three decimal places, we have \[ P(X < 4) \approx 0.382. \] Thus, the probability that fewer than 4 homeowners employ a pool service is \[ \boxed{0.382}. \]