The probability that a mango is medium or large is given by:

\(p = 0.70 + 0.15 = 0.85\)

We need to find the probability that fewer than 12 mangoes are medium or large. This is equivalent to finding:

\(P(X < 12) = 1 - P(X \geq 12)\)

Where \(X\) is the number of medium or large mangoes out of 14. We use the binomial distribution:

\(P(X \geq 12) = P(X = 12) + P(X = 13) + P(X = 14)\)

Using the binomial probability formula:

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

Calculate each probability:

\(P(X = 12) = \binom{14}{12} (0.85)^{12} (0.15)^2\)

\(P(X = 13) = \binom{14}{13} (0.85)^{13} (0.15)^1\)

\(P(X = 14) = \binom{14}{14} (0.85)^{14} (0.15)^0\)

Summing these gives:

\(P(X \geq 12) = 0.6479\)

Thus,

\(P(X < 12) = 1 - 0.6479 = 0.352\)