Jailer’s Paradox

Cousin of the Monty Hall Paradox

Jailer’s Paradox

• If the Monty Hall paradox did not make intuitive sense, perhaps we can look at it from a different perspective.

• This paradox is similar to Monty Hall, but perhaps more intuitive.

Jailer’s Paradox

• Instead of 3 doors, we have 3 prison cells.• You are one of 3 prisoners, 2 of whom will be set

free, and 1 will be put to death.• Your are writing a letter to be delivered to your

family in case you are executed, and you ask the jailer which of the other 2 prisoners could definitely take your letter.

• Would the jailer increase your odds of execution by telling you who to address the letter to?

Jailer’s Paradox

• As a prisoner, should you be more worried after the jailer tells you the name?

Jailer’s Paradox - Solution• Here is the solution from “Randomness”, by Deborah Bennett

Scenario Prisoner’s fate Jailer says this prisoner will go free

Prob- ability of this scenario

Adam Bill Carl

1a Ω Ш Ш Bill 1/6

1b Ω Ш Ш Carl 1/6

2 Ш Ω Ш Carl 1/3

3 Ш Ш Ω Bill 1/3

Simpson’s Paradox

• If the acceptance rate of women is higher than the rate of men in every department in the college, how can the women’s rate be lower than the men’s for the college as a whole?

• From “Randomness”, by Deborah Bennett

Simpson’s ParadoxWomen Men

Dept. Applied Accepted Applied Accepted

1 50 20 (40%) 30 10 (33%)

2 40 30 (75%) 70 50 (71%)

CollegeTotal

90 50 (56%) 100 60 (60%)

From “Randomness”, by Deborah Bennett

Simpson’s Paradox – More Slanted

Star Athletes Star Scholars

Dept. Applied Accepted Applied Accepted

Sports Journalism

30 15 (50%) 1 0 (0%)

Rocket Science

1 1 (100%) 30 20 (67%)

CollegeTotal

31 6 (52%) 31 20 (65%)

Simpson’s Paradox - Revisited

Adapted from “Randomness”, by Deborah Bennett

Women Men Total

Dept. Apply Accept Apply Accept Apply Accept Accept Rate

1 50 20 (40%)

30 10 (33%)

80 30 38%

2 40 30 (75%)

70 50 (71%)

110 80 73%

CollegeTotal

90 50 (56%)

100 60 (60%)