After multiple missiles, the outcomes you care about are "Target was hit" and "Target was not hit". Let's find the probability that after firing missiles, the target was not hit at all. That means every missile missed. The probability for that would be since the probability for each missile is independent. Since there are only two possible outcomes, the probability that the target was hit is therefore . Since you want that probability to be greater than 80%, you set . Solve for .
Edit: If the original post was supposed to show that the missile hits its target with probability 0.3%, then change the equation to .
Solution (if the probability for hitting the target is 0.3=30%):
Take the natural log of both sides:
Since , when we divide both sides by a negative number, it flips the inequality:
, hence, you need at least 5 missiles.
If the probability was 0.3% = 0.003, then change that to , so you need at least 536 missiles.