# A group witnesses a crime. Studies have shown that 1/3 of women tell the truth and 1/2 of men tell the truth. There are 210 people in the group with 120 men; and 14 witnesses are chosen in the same ratio as the number of men and women in the group. More than 50% have to testify that the crime was committed. What is the probability that the culprit goes free.

A group of 210 people consisting of 120 men and 90 women witnesses a crime. Studies have shown that in this situation `1/3` of women and `1/2` of men are likely to tell the truth.

Of the group 14 witnesses consisting of 8 men and 6 women are chosen to testify. If more than 50% of the witnesses testify against the culprit he is arrested.

For the culprit to be convicted the number of men and women that say he committed the crime is either of the following (8,0), (7,1), (6,2), (5, 3), (4, 4), (3, 5), (2, 6).

The probability that 8 men and 0 women tell the truth is `(1/2)^8*(2/3)^6` . Similarly, a probability can be determined for all the other cases. Adding them gives:
`(1/2)^8*(2/3)^6` +`(1/2)^7*(1/2)*(2/3)^5*(1/3)` +`(1/2)^6*(1/2)^2*(2/3)^4*(1/3)^2` +`(1/2)^5*(1/2)^3*(2/3)^3*(1/3)^3` +`(1/2)^4*(1/2)^4*(2/3)^2*(1/3)^4` +`(1/2)^3*(1/2)^5*(2/3)^1*(1/3)^5` +`(1/2)^2*(1/2)^6*(1/3)^6`

= `(1/2^8)*(1/3^6)*(2^6 + 2^5 + 2^4 + 2^3 + 2^2 + 2 + 1)`

= `127/(2^8*3^6)`

= `127/186624`

The probability that the culprit is arrested based on the true testimony of the witnesses is `127/186624`

Approved by eNotes Editorial Team