Let the probability that a customer rates the logo as good be \(p = 0.42\). The number of customers rating the logo as good follows a binomial distribution \(B(n, p)\) where \(n = 10\).

We need to find the probability that fewer than 8 customers rate the logo as good, i.e., \(P(X < 8)\).

This is equivalent to \(1 - P(X \geq 8)\).

Using the binomial probability formula:

\(P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\)

Calculate \(P(X \geq 8) = P(X = 8) + P(X = 9) + P(X = 10)\).

\(P(X = 8) = \binom{10}{8} (0.42)^8 (0.58)^2\)

\(P(X = 9) = \binom{10}{9} (0.42)^9 (0.58)^1\)

\(P(X = 10) = \binom{10}{10} (0.42)^{10} (0.58)^0\)

Thus, \(P(X \geq 8) = 10C_8 (0.42)^8 (0.58)^2 + 10C_9 (0.42)^9 (0.58)^1 + 10C_{10} (0.42)^{10}\).

Therefore, \(P(X < 8) = 1 - (10C_8 (0.42)^8 (0.58)^2 + 10C_9 (0.42)^9 (0.58)^1 + 10C_{10} (0.42)^{10})\).

Calculating this gives \(P(X < 8) = 0.983\).