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Confess or Not

Probability puzzles require you to weigh all the possibilities and pick the most likely outcome.

 Puzzle ID: #8823 Fun: (2.36) Difficulty: (2.66) Category: Probability Submitted By: lesternoronha1 Corrected By: Winner4600

You and your partner in crime are both arrested and questioned separately. You are offered a chance to confess, in which you agree to testify against your partner, in exchange for all charges being dropped against you, unless he testifies against you also. Your lawyer, whom you trust, says that the evidence against both of you, if neither confesses, is scant and you could expect to take a plea and each serve 3 years. If one implicates the other, the other can expect to serve 20 years. If both implicate each other, you could each expect to serve 10 years. You assume the probability of your partner confessing is p. Your highest priority is to keep yourself out of the pokey, and your secondary motive is to keep your partner out. Specifically, you are indifferent to you serving x years and your partner serving 2x years. At what value of p are you indifferent to confessing and not confessing?

If you confess, there is probability p of two confessions and (1-p) probability that only you implicate your partner. If you don't confess, there is probability p that neither of you implicate each other and a (1-p) probability that only you are implicated. Let s denote one unit of suffering. One year in jail yourself would cause you 2 units of suffering, and one year of your partner serving would cause you 1 unit of suffering.

This shows the total suffering given the four possibilities:

Both of you confess: 10*2+10=30
Your partner confesses and you don't: 2*20+0=40
You confess and your partner doesn't: 0+20=20
Neither of you confesses: 2*3+3=9

So given the probability p of your partner confessing if you confess, the expected, or average, amount of suffering would be 30p + 20(1-p)=10p+20. If you don't confess the expected amount of suffering would be 40p + 9(1-p)=31p+9. To find the indifference point equate the two expressions:
10p+20 = 31p+9
11 = 21p
p=11/21 =~ 52.4%.
Thus if you think your partner's probability of confessing is greater than 11/21, you should confess; otherwise, you should keep your mouth shut.

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