How can you get a probability of |X - 10|>1.8? given X~N (10,4)

Question

It is a normal distribution and the we hae to find the probability but the x minus 10 is a modulus

Answer 1

Mean=10, SD=4.

X-10>1.8, X>11.8 or 10-X>1.8, X<8.2.

Convert to Z inequality: |X-10|>1.8, |X-10|/4>0.45, |Z|>0.45 means Z>0.45 or Z<-0.45.

Z=(11.8-10)/4=1.8/4=0.45. This Z score corresponds to probability of 0.6736 (67.36%). But this probability is for Z<0.45, i.e., X<11.8. So we have to work out 1-0.6736=0.3264 (32.64%).

Z=(8.2-10)/4=-1.8/4=-0.45. This Z score corresponds to probability of 1-0.6736=0.3264 (32.64%).

So for probability of X<8.2 OR X>11.8, we add the two probabilities together: 0.6528 (65.28%).

The easiest way to visualise this is to draw the normal distribution curve roughly and mark the Z=0 point which divides the curve into two equal halves. To the left of zero mark Z=-0.45 and to the right Z=0.45. The area to the right of Z=0.45 is where X>11.8, and the area to the left of Z=-0.45 is where X<8.2. These areas are the same, so we find one area and double it. P(-Z)=1-P(Z) so if Z=-0.45, we use Z=0.45 and subtract from 1. Double this to get the area (probability) expressed in the inequality.