Torpedoes Sinking A Ship

Discrete Mathematics Level 2

Suppose that each torpedo has a 1/3 probability of sinking a ship, 1/3 probability of damaging it and 1/3 of missing it altogether. Moreover, 2 damaging shots will also sink the ship. The probability that firing four torpedoes will sink the ship can be expressed as \( \frac{a}{b} \), where \(a\) and \(b\) are coprime, positive integers. Calculate \(a+b\).

Note: All that we care about is that the ship is sunk at the end of the 4 shots. The first shot could already have sunk the ship.

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