SOLUTION: Just when I thought I was begninng to understand probability I came across this problem. I'm really struggling with what "N" is. A manufacturer of valves admits that its quality

Hi, there--

THE PROBLEM:
A manufacturer of valves admits that its quality control has gone radically “downhill” such 
that currently the probability of producing a defective valve is 0.50. If it manufactures 
1 million valves in a month and you randomly sample from these valves 10,000 samples, 
each composed of 15 valves.

a. In how many samples would you expect to find exactly 13 good valves?
b. In how many samples would you expect to find at least 13 good valves?

A SOLUTION:
I agree that it's sometimes difficult to figure out what variables and values to use. 
I would treat this like a Binomial Distribution with X ~ Binom (15, 0.50)
n = sample size = 15
p = probability that randomly selected valve is good (not defective) = 0.50


a. In how many samples would you expect to find exactly 13 good values?

P(X=13) = (15C13) * (0.50)^13* (0.50)^2 = 105 * 0.000122 * 0.25 = 0.0032043

So the probability of finding exactly 13 good valves in your 15-valve sample is
0.0032043457. So, out of 10,000 samples of 15 valves, we'd expect to see 32 samples with exactly 13 good values.

b. In home many samples would you expect to find at least 13 good valves?

P(X≥13) = P(X=13) + P(X=14) + P(X=15)

By the same method:
P(X=14) =0.0004578
P(X=15) =0.0000305

P(X≥13) = 0.0032043 + 0.0004578 + 0.0000305 = 0.0036926

So, out of 10,000 such samples, we expect to see 37 samples with at least 13 good valves.

Hope this helps!

Mrs. Figgy
math.in.the.vortex@gmail.com