After flying between two black holes, our engines have been a bit janky. The probability of one of any of our 8 power couplings failing is 10%: What's the probability none fail? What's the probability one fails? What's the probability four fail? What's the probability six fail? What's the probability eight fail?

Step-by-step explanation:

We would apply the formula for binomial distribution. It is expressed as

P (x = r) = nCr * q^ (n - r) * p^r

Where

p represents probability of success

q represents probability of failure.

n represents number of sample.

From the information given,

p = 10% = 10/100 = 0.1

q = 1 - q = 1 - 0.1 = 0.9

n = 8

1) The probability that none fails is

P (x = 0) = 8C0 * 0.9^ (8 - 0) * 0.1^0

P (x = 0) = 0.43

2) The probability that one fails is

P (x = 1) = 8C1 * 0.9^ (8 - 1) * 0.1^1

P (x = 1) = 0.38

3) The probability that four fails is

P (x = 4) = 8C4 * 0.9^ (8 - 4) * 0.1^4

P (x = 4) = 0.0046

3) The probability that six fails is

P (x = 6) = 8C6 * 0.9^ (8 - 6) * 0.1^6

P (x = 6) = 0.00002268

4) The probability that eight fails is

P (x = 8) = 8C8 * 0.1^ (8 - 8) * 0.1^8

P (x = 8) = 0.00000001