We need to find \(P(Y < 2\mu)\). To do this, we standardize the variable using the formula for the standard normal distribution:

\(z = \frac{Y - \mu}{\sigma}\)

Substituting \(Y = 2\mu\), \(\mu\), and \(\sigma = \frac{2}{3} \mu\), we have:

\(z = \frac{2\mu - \mu}{\frac{2}{3} \mu} = \frac{\mu}{\frac{2}{3} \mu} = \frac{3}{2} = 1.5\)

Thus, \(P(Y < 2\mu) = P(z < 1.5)\).

Using standard normal distribution tables or a calculator, \(P(z < 1.5) = 0.933\).