Stats Probability

Question

A jury has 12 jurors. A vote of at least 10 of 12 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 anyone juror makes the correct decision on a defendant is .80. If the defendant is guilty, what is the probability that the jury makes the correct decision?

Answer 1

each juror makes a correct decision .8 of the time, 80%= 4/5 of the time

12 jurors, 10 needed for conviction, each has an independent chance of voting correctly for guilty= 80%

3 ways to get a conviction. unanimous, 12-0, or slightly less than unanimous: 11-1 or 10-2

12C12(.8)^12(.2)^0 + 12C11(.8)^11(.2)^1 + 12C10(.8)^10(.2)^2

= (.8)^12 + 12(.8)^11).02 + 12(11)(.8^10)(.2)^2

= ..0687194767 + .2061584302 + .5669356831= .84181359

= about 84.18 % chance of an accurate guilty conviction

about 16% chance of an innocent person convicted