probability of a successful launch?

Question

A company is launching rockets, 5 trials in 3 stages. The probaility of success for the first stage is 0.93, for the second stage is 0.93 and the third stage is 0.86.
Questions:
1. what is the probabiliy of a successful first launch
2. what is the probability of the first two launches being failures and the last three successes?

Answer 1

Dear Clecle,

1. If you really mean what you wrote, then the answer to question 1 is 0.86 because if the third stage is successful then the first two stages must have also been successful. Perhaps you, or the author of the problem, intended that the listed probabilities for success of the second and third stages be conditioned on the success of the earlier stages, but that is NOT what you have stated. Instead you have given unconditional probabilities. This is an important distinction to understand when solving (or writing) probability problems.

IF you had conditional probabilities, given the previous stages were successful, then you would need to multiply them together to get the probability of a successful launch.

P(stage 1 success & stage 2 success)
= P(stage 1 success) P(stage 2 success | stage 1 success)
= (0.93) (0.93)
= 0.8649 .

P(stage 1 success & stage 2 success & stage 3 success)
= P(stage 1 success & stage 2 success) P(stage 3 success | stage 1 success & stage 2 success)
= (0.8649) (0.86)
= 0.743814,
which would be the probability that the first launch succeeds IF your probabilities were conditional as described, since all three stages are successful here.


2. Given we don't learn anything new or make changes from launch to launch, we'll treat the probabilities of successful launches independently for each of the five trials. Thus, if s is the probability of a success and f is the probability of a failure, then the probability of the first two launches being failures and the last three successes is given as

(f^2) (s^3),

where you can substitute in 0.86 or 0.743814 for s, depending on whether the problem statement is correct as written or as I suspect the author meant, and the value for f is simply 1 - s.