A certain airplane has two independent alternators to provide electrical power. Whats the probability?

Question

A certain airplane has two independent alternators to provide electrical power. The probability that a given alternator will fail on a 1-hour flight is .02. What is the probability that

Both will fail?

Neither will fail?

One or other will fail?

Answer 1

Yes, but two independent alternators never work at the same. This affects probability of failure.

The failure rate is higher due to the "switch-over effect".

Answer 2

a) .02*.02 = .0004

b) .98 * .98 = .9604

c) 1 - .98 * .98 = .0396

Strange -- it leaves out the most important -- what's the probability that at least one will survive:

1 - .02 *.02 = .9996

Answer 3

Let X be the number of alternators that fail.  X has the binomial distribution with n = 2 trials and success probability p = 0.02



In general, if X has the binomial distribution with n trials and a success probability of p then

P[X = x] = n!/(x!(n-x)!) * p^x * (1-p)^(n-x)

for values of x = 0, 1, 2, ..., n

P[X = x] = 0 for any other value of x.

The probability mass function is derived by looking at the number of combination of x objects chosen from n objects and then a total of x success and n - x failures.

Or, in other words, the binomial is the sum of n independent and identically distributed Bernoulli trials.

X ~ Binomial( n = 2 , p = 0.02 )

the mean of the binomial distribution is n * p = 0.04

the variance of the binomial distribution is n * p * (1 - p) = 0.0392

the standard deviation is the square root of the variance = √ ( n * p * (1 - p)) = 0.1979899

The Probability Mass Function, PMF,

f(X) = P(X = x) is:

P( X =  0 ) =  0.9604 << neither fail.

P( X =  1 ) =  0.0392 << one or the other fails

P( X =  2 ) =  4e-04 << Both fail