The probability of hitting the target is $\frac{3}{5}$ for each shot. The shooter stops firing after the first hit. Find expected value and variance of the number of shots taken. There are exactly four bullets.
Random variable $\xi = $ number of shots taken. $\xi \in \{ 1, 2, 3, 4\}$.
\begin{array} {|r|r|}\hline a & 1 & 2 & 3 & 4 \\ \hline P(\xi = a) & \frac{3}{5} & \frac{2}{5} \cdot \frac{3}{5} & \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{3}{5} & \frac{2}{5}\cdot \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{3}{5} + \left( \frac{2}{5} \right) ^4\\ \hline \end{array}
The sum of probabilities is $1$. $$E\xi = \sum_{i=1}^4a_i \ P(\xi =a_i) = \frac{3}{5} + 2\cdot\frac{2}{5}\cdot\frac{3}{5} + 3\cdot\frac{2}{5}\cdot\frac{2}{5}\cdot\frac{3}{5} + 4\cdot\frac{2}{5}\cdot\frac{2}{5}\cdot\frac{2}{5}\cdot\frac{3}{5} + 4\cdot \left( \frac{2}{5} \right)^4 = 1.624.$$
The textbook answer is $1.52$.
What am I doing wrong?