Question

n. A jury consists of 12 members, and each member must vote either guilty or not guilty. Assume that the vote of each member is independent Suppose that the probability that a juror votes correctly (i.e. in line with the actual innocence/guilt of the person on trial) is 70% What is the probability that there is an inconclusive (tied) voting outcome

Answer 1

0.0792 = 7.92% probability that there is an inconclusive (tied) voting outcome

Explanation:

For each juror, there are only two possible outcomes. Either they vote correctly, or they do not. The probability of a juror voting correctly is independent of any other juror. This means that the binomial probability distribution is used to solve this question.

Binomial probability distribution

The binomial probability is the probability of exactly x successes on n repeated trials, and X can only have two outcomes.

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

In which
`C_(n,x)` is the number of different combinations of x objects from a set of n elements, given by the following formula.

`C_(n,x) = (n!)/(x!(n-x)!)`

And p is the probability of X happening.

12 members

This means that
`n = 12`

Suppose that the probability that a juror votes correctly (i.e. in line with the actual innocence/guilt of the person on trial) is 70%

This means that
`p = 0.7`

What is the probability that there is an inconclusive (tied) voting outcome?

Same number(12/2 = 6) of votes for each option, so P(X = 6).

`P(X = x) = C_(n,x).p^(x).(1-p)^(n-x)`

`P(X = 6) = C_(12,6).(0.7)^(6).(0.3)^(6) = 0.0792`

0.0792 = 7.92% probability that there is an inconclusive (tied) voting outcome