Question

A jury has 13 jurors. A vote of atleast 11 of 13 for "guilty" is necessary for a defendant to be convicted of a crime. Assume that each juror acts independently of the others and that the probability that any one juror makes the correct decision on a defendant is 0.870. If the defendant is guilty, what is the probability that the jury makes the correct decision?

Answer 1

ick

Answer 2

i guess we can compute the probability that 13 vote to convict, 12 vote to convict and 11 vote to convict and add them up

Answer 3

the probability that all thirteen vote to convict is \(0.870^{13}\)

Answer 4

the probability that exactly 12 vote to convict is
\[13\times (.87)^{12}\times .13\]

Answer 5

and finally the probability that exactly 11 vote to convict is
\[\binom{13}{2}(.87)^{11}\times (.13)^2\]

Answer 6

it is all binomial
\[P(x=k)=\binom{n}{k}p^k(1-p)^{n-k}\]

Answer 7

compute all that mess, add them up