From charlesreid1

Friday Morning Math Problem

The Flippant Juror

A 3-person jury has 2 members, each of whom has an independent probability P of making a correct decision. The third juror flips a coin. (Majority rules.)

A 1-person jury has 1 member who has a probability P of making the correct decision.

Which jury has a better probability of making the correct decision?

The probability of a 3-person jury making a correct decision is the union of two situations: Jurors 1 and 3 being correct, or Jurors 2 and 3 being correct:
P_{correct verdict} = ( P_{juror 1 correct} and P_{juror 3 correct} ) or (P_{juror 2 correct} and P_{juror 3 correct})

Juror 1 and 2 have probability P of being right, while Juror 3 has a 50% probability of being right. Furthermore, we know that "and" translates to multiplication of probabilities. We also know that "or" translates to addition of probabilities. The above expression becomes:

P_{3 correct} = 0.5*P + 0.5*P = P

Meanwhile, the 1 person jury has a probability of making a correct decision of:

P_{1 correct} = P
From this, we can see that the two juries have an equal probability of being correct.