The problem involves a binomial distribution where the probability of success (a household having a printer) is 0.68, and the number of trials is 8. We need to find the probability that 5, 6, or 7 households have a printer.
The probability of exactly k successes in a binomial distribution is given by:
\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)
where \(n = 8\), \(p = 0.68\), and \(1-p = 0.32\).
Calculate the probabilities for 5, 6, and 7 successes:
\(P(X = 5) = \binom{8}{5} (0.68)^5 (0.32)^3\)
\(P(X = 6) = \binom{8}{6} (0.68)^6 (0.32)^2\)
\(P(X = 7) = \binom{8}{7} (0.68)^7 (0.32)^1\)
Sum these probabilities:
\(P(5, 6, 7) = \binom{8}{5} (0.68)^5 (0.32)^3 + \binom{8}{6} (0.68)^6 (0.32)^2 + \binom{8}{7} (0.68)^7 (0.32)\)
Using the binomial coefficients and calculations:
\(P(5, 6, 7) = 0.722\)
