Submarine probability problem

Question

A submarine launches 4 missiles to a battleship. The probability that a missile hits is 0.3. For sinking the battleship it is enough that 2 missiles hit it, but if a single missile hits it, the probability that it sinks is 0.6.
What is the probability that the battleship sinks?

I tried to split it into two cases:
A) Only one missile of four hits the battleship: then we get
$P$(sinks) = $P$(missile hits)*$P$(battleship sinks given that it was hit by a missile) = 0.3*0.6 = 0.18. Of course we multiply this probability with $\binom{4}{1}$ for choosing only one missile that hits the target.

B) Two missiles hit he target
$P$(sinks) = P(missile hits)*$P$(missile hits) = 0.09 and of course we multipy this value with $\binom{4}{2}$, same reasoning as above.

In the end, I added A and B and that should be the final prbability.
Do you have any idea if it is correct?

Answer 1

Your answer is not correct, because you did not take into account that while some missiles hit, others have to miss. We can distinguish two cases in which the ship does not sink:

1. Only one missile hits and this missile is not crucial, with probability ${4 \choose 1} \cdot 0.3 \cdot 0.7^3 \cdot 0.4$

2. No missiles hit, with probability $0.7^4$

The probability of the ship sinking thus equals:

$$1 - 4 \cdot 0.3 \cdot 0.7^3 \cdot 0.4 - 0.7^4 = 0.59526$$

Answer 2

No the given solution is wrong. In your first case when you are assuming only one missile is hitting, this means that the rest 3 are NOT hitting. You have not considered their probability, So the first case would actually be,
P(sinks) = $\binom{4}{1}\times0.3\times(0.7)^3\times0.6$

Similarly do the case B.
Also submarine will sink if 3 or 4 missiles also hit, you haven't considered that case. Final answer would be
P(sinks) = $\binom{4}{1}\times0.3\times0.7^3\times0.6 + \binom{4}{2}\times0.3^2\times0.7^2 + \binom{4}{3}\times0.3^3\times0.7 + \binom{4}{4}\times0.3^4$

Answer 3

Your solution is not okay (see the answer of e.g. jvdhooft).

General idea of solving this:

Let $S$ denotes the event that the ship sinks and let $X$ denote the number of missiles that hit the ship.

Then:

$$P\left(S\right)=\sum_{i=0}^{4}P\left(X=i\right)P\left(S\mid X=i\right)\tag1$$

Now substitute the (known values) of $P(X=i)$ and $P(S\mid X=i)$.

Of course you can also use $(1)$ with $S$ replaced by $S^{\complement}$ and then take advantage of $P(S)=1-P(S^{\complement})$.