Maths Question- An aircraft is designed to operate with 99.99% reliability over 100hrs of flight.?

Question

Here's the question- How long would the aircraft need to operate for the chance of continuous successful operation to reduce to 50%? I'm really more interested HOW the problem is solved, not the answer.

Answer 1

I'm not too sure if this is the correct way, but I'll give it a shot.

Consider the problem in terms of binomial distribution, because each block of 100 hours of flight can be considered independent events (i.e. not conditional probability).

Pr(success)= 0.9999

Pr(failure )=0.0001

Number of trials=x

Number of successes=x

Therefore;

nCr(x,x) * [Pr(success)]^(x-0) * [Pr(failure)]^(x-x)=0.5

[Pr(success)]^x=0.5

0.9999^x=0.5

x= 6931.1252 (to 4 decimal places)

This means 6931.1252 sets of 100 hour blocks, translating into approximately 693113 hours.

Another way to think of this is to visualize a tree diagram. You have two branches, which split off into a further two branches, etc... If the top branch represents the probability of success (i.e. probability is 0.9999) and the bottom branch represents failure, how many times do you have to go along the top branch before you reach a probability of 0.5 (i.e. 0.9999^x=0.5, solve for x)? Remember that when you go along a branch, you multiply, and when you go across branches you add.

By the way, in case it's not clear, nCr(n,m) means 'n choose m', or the number of ways 'n' objects can be chosen from a total of 'm' objects.