Thread: Conditional expectation of a PMF

I'm having trouble starting this problem.

Say you committed a crime and you're sentenced to jail. When entering the jail, you pick one ball from a box containing 3 balls, each numbered 0, 1, and 3. If you select 0, you get out of jail; if you select 1 or 3, you put the ball back into the box and after that number of years, you select again under the same conditions. You repeat this until you select a 0 and you're free to go. What is the expected value of how long you'll be in jail for?

The hint is that you're suppose to condition on the first ball selected. I don't see how this helps?

Set E as the expected time to be in jail. Now there is three cases:

1 (1/3 probability): 0 years
2 (1/3 probability): 1 year, then you will have to do the same thing again, so in average 1 + E
3 (1/3 probability): 3 years, then you will have to do the same thing again, so in average 3 + E

To add it up, E = (1/3)*0 + (1/3)*(1+E) + (1/3)*(3+E). Now this is a regular linear equation.

Originally Posted by TriKri

Set E as the expected time to be in jail. Now there is three cases:

1 (1/3 probability): 0 years
2 (1/3 probability): 1 year, then you will have to do the same thing again, so in average 1 + E
3 (1/3 probability): 3 years, then you will have to do the same thing again, so in average 3 + E

To add it up, E = (1/3)*0 + (1/3)*(1+E) + (1/3)*(3+E). Now this is a regular linear equation.

Solving that equation for E, I get E = 11 years.

I don't understand why for 1 year, the average would be 1 + E. Is it because after 1 year, you select again and the expectation of selecting then is the same as selecting now?

I think E = 4, try to resolve the linear equation again.

Originally Posted by CUEngineering

Solving that equation for E, I get E = 11 years.

E = 11 doesn't solve the equation. Try to solve it again; it should be 4, as impossibletask wrote.

Originally Posted by CUEngineering

I don't understand why for 1 year, the average would be 1 + E. Is it because after 1 year, you select again and the expectation of selecting then is the same as selecting now?

Yes, that's exactly why.