The probability that a motor will malfunction within 5 years of manufacture is \(0.03\). Out of eight motors calculate the probability that within 5 years of manufacture (a) all eight will malfunction (b) six will malfunction (c) none will malfunction.

For each scenario, calculate the binomial coefficient C(n, k). We have n = 8, and we'll find the coefficients for k = 0, 6, and 8. (a) C(8, 8) = \(\frac{8!}{8!(8-8)!}\) = 1 (b) C(8, 6) = \(\frac{8!}{6!(8-6)!}\) = \(\frac{8*7}{2}\) = 28 (c) C(8, 0) = \(\frac{8!}{0!(8-0)!}\) = 1

Now, we will use the binomial probability formula to calculate the probabilities required for each scenario. (a) all eight will malfunction: P(X=8) = C(8, 8) * (0.03)^8 * (1-0.03)^(8-8) P(X=8) = 1 * (0.03)^8 * (1)^0 P(X=8) = 6.57*10^(-5) (b) six will malfunction: P(X=6) = C(8, 6) * (0.03)^6 * (1-0.03)^(8-6) P(X=6) = 28 * (0.03)^6 * (0.97)^2 P(X=6) = 2.39*10^(-4) (c) none will malfunction. P(X=0) = C(8, 0) * (0.03)^0 * (1-0.03)^(8-0) P(X=0) = 1 * (1) * (0.97)^8 P(X=0) = 0.738 The probabilities are as follows: (a) the probability that all eight motors will malfunction is approximately 6.57*10^(-5), (b) the probability that six motors will malfunction is approximately 2.39*10^(-4), and (c) the probability that none of the motors will malfunction is approximately 0.738.