# Statistics - normal distribution

cohkka

## Homework Statement

A plant manufactures 500 components a day with the diameter being random variable:
N(8.02, 0.1^2)mm

What is the probability of two randomly picked components differing by more than 0.3mm?

2. Solution
I know that the solution is 0.966

## The Attempt at a Solution

I thought that I needed to find the probability of the diameter differing from the mean by more than 0.3mm:
P(X < 7.72) + P(X > 8.32)
= 1 - P(7.72 < X < 8.32)
= 1 - { Phi[(8.32-8.02)/0.1] - Phi[(7.72-8.02)/0.1] }
= 1 - { Phi - Phi[-3] }
= 1 - { Phi - (1 - Phi) }
= 1 - {0.99865 - (1-0.99865)}
= 1 - 0.9973
= 0.0027

This obviously is far from the correct answer and so I realise I must be completely on the wrong track but don't know how else one might approach this problem. I thought that the question might instead be asking for the probability of the difference of the two diameters being greater than 0.3mm so
P(a < X < a+0.3) but have no idea how you would go about finding the probability of an unknown value.

As you can probably tell, I am really confused by this so any help would be much appreciated! Thank you!

Staff Emeritus
Homework Helper
You have two random variables: X1, the diameter of the first ball, and X2, the diameter of the second ball. Now form a third random variable Y=X1-X2, which is the difference in their diameters. Do you see how to solve the problem now?

cohkka
I think I might have got it now...

So for this new variable Y=X1-X2, I have to also calculate a new standard deviation which is sqrt(0.12 + 0.12) = 0.1*sqrt(2)

So P(X>0.3) = 1 - P(X<0.3)
= 1 - Phi[0.3/(0.1*sqrt(2))]
= 1 - Phi[2.1213...]
= 1 - 0.983
= 0.0169
P(X<-0.3) = 0.0169
P(-0.3<X<0.3) = 1 - 2*0.0169
= 0.9661 (I guess the solution must have just been given for the difference being less than 0.3mm??)

Is this right? I understand why this would be correct but then when I apply this to the next part of the question, I do not get the right answer.
Now another plant B produces components with diameter being random variable N(7.95,0.082)

What now is the probability of the diameter of two randomly picked components differing by more than 0.3mm if one is produced by plant B and the other by the first plant?

sqrt(0.12 + 0.082) = [sqrt(41)]/50

Using the mean as zero and this standard deviation, I get P(-0.3<X<0.3)=0.9808, whereas the solution I am given is 0.9662.....where am I going wrong

Sorry for the long post!

Staff Emeritus