A person tried by a 3 -judge panel is declared guilty if at...

A person tried by a 3 -judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability $.7,$ whereas when the defendant is in fact innocent, this probability drops to $.2 .$ If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that (a) judges 1 and 2 vote guilty; (b) judges 1 and 2 cast 1 guilty and 1 not guilty vote; (c) judges 1 and 2 both cast not guilty votes. Let $E_{i}, i=1,2,3$ denote the event that judge $i$ casts a guilty vote. Are these events independent? Are they conditionally independent? Explain.

A person tried by a 3 -judge panel is declared guilty if at least 2 judges cast votes of guilty. Suppose that when the defendant is in fact guilty, each judge will independently vote guilty with probability $.7,$ whereas when the defendant is in fact innocent, this probability drops to
$.2 .$ If 70 percent of defendants are guilty, compute the conditional probability that judge number 3 votes guilty given that
(a) judges 1 and 2 vote guilty;
(b) judges 1 and 2 cast 1 guilty and 1 not guilty vote;
(c) judges 1 and 2 both cast not guilty votes.
Let $E_{i}, i=1,2,3$ denote the event that judge $i$ casts a guilty vote. Are these events independent? Are they conditionally independent? Explain.

Key Concepts

Conditional Independence

Conditional independence refers to a situation where two events are independent given the occurrence of a third event. In such cases, once the third event is known, information about one of the events does not provide further information about the other. This concept is particularly significant in models where a common factor influences several outcomes, making those outcomes unconditionally dependent but independent when conditioning on the common factor.

Independence

Independence in probability theory means that the occurrence of one event does not affect the probability of another event occurring. When events are independent, the joint probability of the events equals the product of their individual probabilities. Understanding independence is crucial in simplifying the calculation of complex probabilities when events do not influence each other.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has occurred. It is a fundamental concept in probability theory that allows us to revise probabilities based on new information. In problems involving multiple events, such as votes or decisions, conditional probability helps us compute the likelihood of one event in the backdrop of already known outcomes of other events.

Law of Total Probability

The law of total probability is a rule that decomposes the probability of an event into a weighted sum of probabilities conditional on a partition of the sample space. This concept is useful when the overall probability of an event depends on several underlying conditions or cases, each with their own probabilities. It helps in organizing and computing probabilities systematically when multiple scenarios are present.

Bayes' Theorem

Bayes' Theorem provides a way to update prior beliefs based on new evidence. It relates the conditional and marginal probabilities of events, allowing for the inversion of conditional probabilities. This theorem is particularly important in scenarios where information about one event affects the likelihood of another, enabling rational decision making under uncertainty.

Transcript

00:01 In problem 46, it takes at least 9 votes out of 12 of jury members to convict a defendant.

00:12 And we know that a jury member has a probability to vote a guilty person to be innocent is 0 .2.

00:21 Then we have a guilty person.

00:25 It has a probability of 0 .2 to be declared as innocent by a jury member.

00:33 To be innocent and of course it has 0 .8 % to be announced as guilty or voted to be a guilty by a jury member and we have an innocent person can be voted as guilty person by a probability of 0 .1 and it can be declared as an innocent by a remember and we know that each juror act is independently and there a 65 % of the defendants are guilty then for the defendants we have 65 % of them or let's work in decimals we have 0 .65 probability to be guilty and of course 0 .35 % to be innocent this is for the defendants.

01:56 We want to find the probability that the jury renders a correct decision.

02:04 In the first of the problem, the first part of the problem, we know that at least nine out of 10, 9 out of 12 are needed to declare or convict a defendant.

02:18 Let's see the probability for a jury member to vote a person to be guilty.

02:27 We have here a 0 .8 percentage to be guilty and we have here another 0 .1 and the probability to have a guilty person from the defendant is 0 .65 which means from 0 .65 we have two probabilities it can be declared guilty by a percentage of 0 .8 then 0 .8 multiplied by 0 .65 gives 0 .6 0 .52 to be declared as guilty and we have 0 .2 probability to be declared as innocent or be voted by a jury member as innocent then it's 0 .2 multiplied by 0 .65 it's 0 .13 to be voted as innocent and we have here the same the other 0 .35 % to be innocent.

03:34 It can be mistaken voted as guilty by 0 .1 % then 0 .1 multiplied of 0 .35 is 0 .035 can be declared as guilty and we have another 0 .9 % of 0 .35 to be voted as innocent or a correct decision 0 .315 as innocent...

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