Probability - combinations

lalous

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Before you turn on a nuclear power plant, it is subjected to a first roundstructure of three independent tests. Each test has a probability of failure p. Followingperforming these tests
If all three tests fail, then we refuse its application function;
If one or both tests fail, then again the three tests;
If we get three wins, then a second run is performed three testssecurity.

The three tests of the second round are also independent and each a probabilityfailure (p*M). Following the implementation of the second test round
If we get three wins, then it authorizes the function of the plant;
If one or both tests fail, then we start the second round of tests;
If all three tests fail, then we start the procedure in the first inningtests.

There is N nuclear power plants to be commissioned. What is the maximum valuep so that the probability of accepting at least central L is greater than W?

Im trying to do te combination, but te fact that there is 2 rounds of tests makes it too hard for me to understand
 
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Before you turn on a nuclear power plant, it is subjected to a first roundstructure of three independent tests. Each test has a probability of failure p. Followingperforming these tests
If all three tests fail, then we refuse its application function;
If one or both tests fail, then again the three tests;
If we get three wins, then a second run is performed three testssecurity.

The three tests of the second round are also independent and each a probabilityfailure (p*M). Following the implementation of the second test round
If we get three wins, then it authorizes the function of the plant;
If one or both tests fail, then we start the second round of tests;
If all three tests fail, then we start the procedure in the first inningtests.

There is N nuclear power plants to be commissioned. What is the maximum valuep so that the probability of accepting at least central L is greater than W?

Im trying to do te combination, but te fact that there is 2 rounds of tests makes it too hard for me to understand

What are your thoughts?

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Ok here is what I tried to do:

First i tought about the Binomial law, and i tried to solve it backward

P(L>2) = 1 - P(L=0) - P(L=1) = 1 - p^8 - 8(1-p)^7 < 0.89

i took the 2 real roots, and took the one that is between 0 and 1. but then im stuck


The fact that there can be an infnit loop makes it harder for me. I think i need to find the probability of success after ''n'' tests. Idk how to do that
 
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