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 expected (mean) number of years you will spend in jail?

Answer 1

ANSWER:

• present X be the number chosen by the judge, which is the number of years to spend in jail.
• at the present here are 10+5+1=16 numbers in sum of which here are ten 1-year sentences, five 2-year sentences and one 20-year sentence.
• after that the probability of select 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.

Therefore,

P(X=1)=10/16,

P(X=2)=5/16

and

P(X=20)=1/16

• after that 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

though,

• Difference of the predictable number of years to spend in jail is zero because the expected number is a stable.