Question

Upon entering the jail, you draw a ball from a box containing 3 balls numbered 0, 1 and 3 respectively. If you draw the ball numbered 0, you get out of jail immediately. If you draw 1 or 3, you put the ball back in the box and serve that many years in jail, at which time you draw again. This is repeated, until you draw the zero and go free. How long do you expect to be in jail? Hint: condition on the first ball drawn

Answer 1

here let expexcted time in jail =X

hence P(X=0) =P(getting a 0)=1/3

P(X=1+E(X))=P(getting a 1) =1/3

P(X=3+E(X))=P(getting a 3)=1/3

hence E(X)=xP(x)=0*(1/3)+(1+E(X))*(1/3)+(3+E(X))*(1/3)

3*E(X)=4+2E(X)

E(X)=4 YEars