Question

Length of wands from Ollivander’s wand shop is a normally distributed random variable with mean 12 in. and variance 2.25 in2 . What is the probability that a randomly selected wand from the shop is: (a) between 8.25 in and 13.875 in. (b) longer than 8.25 in. (c) shorter than 12 in.

Answer 1

Given,

= 12, = sqrt(2.25) = 1.5

We convert this to standard normal as

P(X < x) = P( Z < x - / )

a)

P( 8.25 < X < 13.875) = P( X < 13.875) - P( X < 8.25)

= P( Z < 13.875 - 12 / 1.5) - P( Z < 8.25 - 12 / 1.5)

= P( Z < 1.25) - P( Z < -2.5)

= P( Z < 1.25) - ( 1 - P( Z< 2.5) )

= 0.8944 - ( 1 - 0.9938)

= 0.8882

b)

P( X > 8.25) = P( Z > 8.25 - 12 / 1.5)

= P( Z > -2.5)

= P( Z < 2.5)

= 0.9938

c)

P( X < 12) = P( Z < 12 - 12 / 1.5)

= P( Z < 0)

= 0.5