Question

What is the probability that you select a vial that is within the acceptable range( in other words you do not detect that the machine shifted)

You randomly select 15 vials what is the probability that you select at least one vial that is within acceptable range? (Acceptable range 9.65 and 9.95)

I do have the answer (0.9988)

I don't understand what formula is being used. How do I solve this and what formula is being used to solve it?

Answer 1

Lets assume that the vials are randomly distributed with mean = 10 and standard deviation = 0.05.

Converting vials to Normal distribution with mean 0 and standard deviation 1 will be done with the help of Z scores

Z = X - mean / standard deviation

Now,

P (9.65 < X < 9.95) = P (7 < Z < 1) (transformed from ab0ve formula)

= P (Z > 7) - P( Z < 1)

= 0 - (-0.15) [ Look up the values from Z table)

= ~ 0.15

P.S - the whole idea of converting to normal distribution of mean 0 and std 1 and z score is to find probabilities from z table

probability that all 15 vials are outside acceptable range = (1 - 0.15)¹⁵

= 0.08

probability that you select at least one vial that is within acceptable range = 1 - 0.08 = 0.92

Substitute assumed mean of 10 with actual mean and you will get your exact answer (0.9988)