Let the random variable \(X\) represent the number of houses with solar heating. \(X\) follows a binomial distribution with parameters \(n = 19\) and \(p = 0.12\), i.e., \(X \sim B(19, 0.12)\).
We need to find \(P(X < 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)\).
Using the binomial probability formula \(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\), we calculate:
\(P(X = 0) = \binom{19}{0} (0.12)^0 (0.88)^{19} = (0.88)^{19}\)
\(P(X = 1) = \binom{19}{1} (0.12)^1 (0.88)^{18} = 19 \times (0.12) \times (0.88)^{18}\)
\(P(X = 2) = \binom{19}{2} (0.12)^2 (0.88)^{17} = \frac{19 \times 18}{2} \times (0.12)^2 \times (0.88)^{17}\)
\(P(X = 3) = \binom{19}{3} (0.12)^3 (0.88)^{16} = \frac{19 \times 18 \times 17}{6} \times (0.12)^3 \times (0.88)^{16}\)
Summing these probabilities gives:
\(P(X < 4) = (0.88)^{19} + 19 \times (0.12) \times (0.88)^{18} + \frac{19 \times 18}{2} \times (0.12)^2 \times (0.88)^{17} + \frac{19 \times 18 \times 17}{6} \times (0.12)^3 \times (0.88)^{16}\)
Calculating these values, we find \(P(X < 4) = 0.813\).
