Consider an m-member jury that requires n or more votes to convict a defendant. Let p...

Consider an m-member jury that requires n or more votes to convict a defendant. Let p be the probability that a juror votes a guilty person innocent and let q be the probability that a juror votes an innocent person guilty, 0<p<1, 0<q<1. Assuming that r is the fraction of guilty defendants and that jurors vote independently, what is

(a) the probability a defendant is convicted?

(b) the probability a defendant is convicted when n=9, m=12, p=.25, q=.2, and r=5/6? Use R to calculate the result.

Solutions

Expert Solution

(a)

Let G and I be the events that the defendants is guilty and innocent respectively.

Let C and NC be the event that the defendants is convicted or not convicted.

Probability of guilty defendants, P(G) = r

Probability of innocent defendants P(I) = 1 - r

Probability that a juror votes a guilty person innocent , P(NC | G) = p

Probability that a juror votes an innocent person guilty, P(C | I ) = q

Probability that a juror votes a guilty person innocent , P(C | G) = 1 - P(NC | G) = 1 - p

Probability a defendant is convicted by one judge, P(C) = P(G) P(C | G) + P(I) P(C | I )

= r (1-p) + (1-r) q

Let X be the number of judges who votes to convict a defendant. Then X ~ Binomial(m, r (1-p) + (1-r) q)

Probability a defendant is convicted = P(X n)

where X ~ Binomial(m, r (1-p) + (1-r) q)

(b)

For, n=9, m=12, p=.25, q=.2, and r=5/6

Probability a defendant is convicted by one judge, P(C) = 0.6583333333

use R, pc = (5/6) * (1-0.25) + (1-5/6) * 0.2

Using the function, pbinom(8, 12, 0.6583333333, lower.tail = TRUE)

Probability a defendant is convicted = P(X n) = 0.6304622129

orchestra answered 1 year ago

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