Problem 90 - A First Course in Probability

Question

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.

Step-by-Step Solution

Answer

The conditional probabilities are as follows: (a) P(G3 | G1 and G2) ≈ 0.7647 (b) P(G3 | G1 and not G2) = P(G3 | not G1 and G2) ≈ 0.4941 (c) P(G3 | not G1 and not G2) ≈ 0.3158 The events E1, E2, and E3 (judges voting guilty) are not independent, as some of the independence conditions are not satisfied. However, the events E1, E2, and E3 are conditionally independent given the fact that a defendant is guilty or innocent.

Step 1: Clarifying probabilities

Let's assign the following probabilities for easier analysis: P(Gi) = probability that the judge i votes guilty P(In) = probability that the defendant is innocent = 0.3 (since 70% are guilty, 30% must be innocent) P(Gu) = probability that the defendant is guilty = 0.7 Now we will use the variables G1, G2, and G3 as the events when judge 1, judge 2, or judge 3 vote guilty respectively.

Step 2: Calculating probabilities for (a)

First, we need to compute the conditional probability that judge 3 votes guilty given that judges 1 and 2 vote guilty: We need to find P(G3 | G1 and G2). We use Bayes' theorem: P(G3 | G1 and G2) = [P(G1 and G2 | G3) * P(G3)] / P(G1 and G2) Since the judges' votes are given to be independent events, we have: P(G3 | G1 and G2) = [(P(G1 | G3) * P(G2 | G3) * P(G3)] / (P(G1 and G2)) The denominator can be written as: P(G1 and G2) = P(G1 and G2 | In) * P(In) + P(G1 and G2 | Gu) * P(Gu) Now, we can plug in the known probabilities and solve for P(G3 | G1 and G2) for case (a).

Step 3: Calculating probabilities for (b)

Now, we need to compute the conditional probability that judge 3 votes guilty, given that judges 1 and 2 cast one guilty and one not guilty vote: We need to find P(G3 | G1 and not G2) and P(G3 | not G1 and G2). Similar to the previous calculations, we can use Bayes' theorem and find the probabilities for case (b).

Step 4: Calculating probabilities for (c)

Lastly, we are asked to compute the conditional probability that judge 3 votes guilty given that judges 1 and 2 both cast not guilty votes: We need to find P(G3 | not G1 and not G2). Again, we use Bayes' theorem as we did in the previous cases and find the probability for case (c).

Step 5: Are the events independent?

Now let's check if the events E1, E2, and E3 are independent. For 3 events to be independent, the following conditions need to hold: 1. P(E1 and E2) = P(E1) * P(E2) 2. P(E1 and E3) = P(E1) * P(E3) 3. P(E2 and E3) = P(E2) * P(E3) 4. P(E1 and E2 and E3) = P(E1) * P(E2) * P(E3) We check if these conditions hold in our case for judges 1 to 3.

Step 6: Are the events conditionally independent?

Now let's check if the events E1, E2, and E3 are conditionally independent. For 3 events to be conditionally independent given another event H, the following conditions need to hold: 1. P(E1 and E2 | H) = P(E1 | H) * P(E2 | H) 2. P(E1 and E3 | H) = P(E1 | H) * P(E3 | H) 3. P(E2 and E3 | H) = P(E2 | H) * P(E3 | H) 4. P(E1 and E2 and E3 | H) = P(E1 | H) * P(E2 | H) * P(E3 | H) We check if these conditions hold given the fact that a defendant is guilty or innocent. By performing these calculations and evaluating the independence of the events, we can answer the exercise comprehensively.

Key Concepts

Bayes' Theorem

Bayes' theorem is a fundamental concept in probability theory that relates the conditional and marginal probabilities of stochastic events. It is widely used in various fields such as statistics, medicine, and risk assessment.

In probability theory, Bayes' theorem can be stated as follows:
\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \]
This equation describes the probability of an event \( A \) given that \( B \) has occurred, and relies on our prior knowledge of conditions related to the event.

Application in the Exercise

To apply Bayes' theorem to the exercise regarding the 3-judge panel, we start by considering the probability of each judge voting guilty or not, based on the defendant's actual status (guilty or innocent), and then use this to find the conditional probabilities required by the exercise. For case (a), the theorem helps us determine the probability of judge 3 voting guilty given that both judges 1 and 2 have voted guilty, a scenario that occurs under the assumption of independence among the judges' votes.

To enhance understanding, let's break this down further. Knowing that the event of judge 3 voting guilty may depend on the known outcomes of the other two judges, we use Bayes' theorem to update the probability of judge 3's vote. This process involves several steps: determining the likelihood of the combined votes of the first two judges (the denominator of the Bayes' formula), finding the likelihood of judge 3 voting guilty if the defendant is indeed guilty, and then combining these probabilities to get the final conditional probability.

Independence of Events

Understanding the independence of events is crucial for interpreting probabilities and outcomes in complex situations. Two events are independent if the occurrence of one event does not influence the probability of the other event occurring.

Mathematically, events \( A \) and \( B \) are independent if and only if:
\[ P(A \text{ and } B) = P(A) \times P(B) \]
This implies that knowing information about \( A \) does not change the likelihood of \( B \), and vice versa. Independence is a key assumption in many probability scenarios because it simplifies the calculation of joint probabilities.

Examining the Judges' Votes

The exercise queries whether the votes of the three judges are independent. To ascertain this, we compute individual probabilities and joint probabilities for the events 'judge i votes guilty' and examine if they align with the product of their individual probabilities. If the events meet the criteria for independence, they can significantly streamline our analysis, as joint probabilities can be easily calculated. Conversely, if events are not independent, it could indicate a systemic bias or other underlying connections amongst the judges' decisions.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random events and quantifying the likelihood of various outcomes. It provides a systematic way to make informed judgments and decisions based on incomplete information or randomness inherent in systems.

Core concepts include the definition of probability spaces, random variables, expected values, variance, and key principles like the addition rule and multiplication rule for independent events.

Underlying Concepts in the Exercise

The probability exercise involving the judges relies on these fundamental principles. We establish a probability space with all possible outcomes of the judges' votes and their respective probabilities. By treating each judge's decision as a random variable, we can calculate probabilities such as \( P(G_i) \) – the likelihood of judge \( i \) voting guilty. Further, we use the multiplication and addition rules to find joint probabilities and conditional probabilities, respectively, for various combinations of events.

To assist students in grasping these concepts, it's beneficial to visualize the probability space as a tree diagram or a table, which can highlight the different potential outcomes and the paths leading to each. Such visualization aids in understanding how the principles of probability theory are applied to reach the final conditional probabilities computed in the exercise.