Let the random variable \(X\) represent the number of days the train is late out of 7 days. \(X\) follows a binomial distribution with parameters \(n = 7\) and \(p = 0.35\).

We need to find \(P(X < 3) = P(X = 0) + P(X = 1) + P(X = 2)\).

Using the binomial probability formula:

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

Calculate each probability:

\(P(X = 0) = \binom{7}{0} (0.35)^0 (0.65)^7 = 0.65^7 = 0.049022\)

\(P(X = 1) = \binom{7}{1} (0.35)^1 (0.65)^6 = 7 \times 0.35 \times 0.65^6 = 0.184776\)

\(P(X = 2) = \binom{7}{2} (0.35)^2 (0.65)^5 = 21 \times 0.35^2 \times 0.65^5 = 0.29848\)

Sum these probabilities:

\(P(X < 3) = 0.049022 + 0.184776 + 0.29848 = 0.532\)