Juries in the Law Courts can find a defendant to be Guilty...

Juries in the Law Courts can find a defendant to be Guilty or Not Guilty, irrespective of whether in reality the defendant is guilty or innocent. In the Law Courts, it has long been believed that a defendant who is guilty will be found Guilty by a jury with probability 0.95, and that an innocent defendant will be found Not Guilty with probability 0.95. It is also believed that 96% of all defendants are guilty. 1. With what probability will a jury find a guilty defendant to be Not Guilty? 2. What is the probability that a jury will find a defendant to be Not Guilty? 3. If a defendant has been found Not Guilty by a jury, what is the probability that the defendant really is guilty?

Transcript

00:04 So to answer this question, i think it's really helpful to make a tree diagram and it helps you find the probability.

00:13 So i'm going to demonstrate that.

00:16 So with my tree diagram, start over here at the far left where i just made the red dot.

00:23 So it says that 96 % of people of defendants are guilty and that means 4 % are innocent.

00:33 So remember, when you make a tree diagram, these probabilities here along the branches must add up to one or 100%.

00:43 So if 96 % of defendants are guilty, then that means 4 % are innocent.

00:49 So then now i'm over here on this branch right here.

00:54 So if they're guilty, 95 % of the jury makes the correct decision, 95 % of the time.

01:03 They find them guilty.

01:05 And that means, look at this, these have to add up to one.

01:10 That means five percent of guilty defendants are found innocent.

01:16 So that means the jury made an incorrect decision.

01:20 And if you're down here on the innocent part, if i'm down here, 95 % of the time the jury makes a correct decision and finds them innocent.

01:31 If they're, you know, since they really are innocent, but 5 % of the time, the jury makes an incorrect decision and finds the defendant guilty.

01:41 So notice these add up to 100%.

01:44 So now i'd like you to look at these numbers over here that i'm on the right that i'm kind of highlighting in green.

01:51 So the way i'm getting these numbers is i'm multiplying along the branches.

01:56 So to get the 0 .912, i took the 0 .96 and multiplied by the 0 .95.

02:03 So 0 .96 times 0 .95 is my 0 .912.

02:08 0 .96 times 0 .05 is 0 .08 .04 times 0 .035 is 0 .08.

02:18 And 04 times 0 .05 is 0 .02.

02:24 Notice that these four probabilities i have on the far right add up to 1 or 100%.

02:31 So those are your 4 .05 .02.

02:32 Possible scenarios.

02:35 Either the defendant is guilty and the jury finds them guilty.

02:38 The defendant is guilty and the jury finds them innocent.

02:41 The defendant is innocent and the jury finds them innocent.

02:45 And the defendant is innocent and the jury incorrectly finds them guilty.

02:49 So those are your four scenarios that can happen.

02:53 So now that i have those probabilities, i can go back and answer these questions.

02:58 So it says with what probability will a jury find a guilty defendant to be not guilty.

03:07 So a guilty defendant, you're going to this branch, be not guilty.

03:12 So the answer to question a, here's the answer to question a is 0 .048.

03:22 So let me erase that.

03:26 That looks like a six.

03:29 Well, i've already got it, an arrow drawn there.

03:33 So the question a, the probability that a guilty person, the jury finds innocent is 0 .08 or 4 .8%...

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