The problem can be approximated using a normal distribution because the number of days is large (100 days), and the probability of messaging is not too close to 0 or 1.

First, calculate the mean and variance of the binomial distribution:

Mean: \(n \times p = 100 \times 0.72 = 72\)

Variance: \(n \times p \times (1-p) = 100 \times 0.72 \times 0.28 = 20.16\)

We approximate the binomial distribution with a normal distribution \(N(72, 20.16)\).

To find the probability of fewer than 64 days, use continuity correction: \(P(X < 64) \approx P(X < 63.5)\).

Standardize using the formula:

\(z = \frac{63.5 - 72}{\sqrt{20.16}}\)

\(z = \frac{63.5 - 72}{4.49} \approx -1.893\)

Using standard normal distribution tables or a calculator, find \(P(z < -1.893)\).

The probability is approximately 0.0292.