*Q*_{4} = probability that 4 depth charges miss the submarine

$Q_4 = \left( \frac{1}{4} \right)^4$

$Q_4 = \frac{1}{256}$

*Q*_{3} = probability that 3 depth charges miss the submarine and 1 damage

$Q_3 = \left( \frac{1}{4} \right)^3 \cdot \left( \frac{1}{4} \right)^1 \times {^4}C_3$

$Q_3 = \frac{1}{64}$

*Q*_{escape} = Probability that the submarine will escape in 4 depth charges

$Q_\text{escape} = Q_3 + Q_4$

$Q_\text{escape} = \frac{5}{256}$

*P*_{sunk} = Probability that 4 depth charges will sink the submarine

$P_\text{sunk} = 1 - Q_\text{escape}$

$P_\text{sunk} = \frac{251}{256}$ ← *answer*