Question

Suppose you commit a crime. The way the judge is going to set your sentence is by pulling a number out of a hat. You know that there are ten 1-year sentences, five 2-year sentences and one 20-year sentences in the hat. What is the variance of the expected number of years you will spend in jail?

Answer 1

Let X be the number selected by the judge, which is the number of years to spend in jail.

Now there are 10+5+1=16 numbers in total, of which there are ten 1-year sentences, five 2-year sentences and one 20-year sentence.

Then the probability of selecting a 1-year sentence is 10/16,  probability of selecting a 2-year sentence is 5/16 and  probability of selecting a 20-year sentence is 1/16.

Thus P(X=1)=10/16, P(X=2)=5/16 and P(X=20)=1/16

Then expected number of years to spend in jail=E(X)=1*10/16+2*5/16+20*1/16=2.5

E(X2)=1*10/16+2*2*5/16+20*20*1/16=26.875

Var(X)=26.875-(2.5)*(2.5)=20.625
However,  variance of the expected number of years to spend in jail is zero as the expected number is a constant.

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