Question

A sample is selected from one of two populations, \(S_{1}\) and \(S_{2},\) with \(P\left(S_{1}\right)=.7\) and \(P\left(S_{2}\right)=.3 .\) The probabilities that an event A occurs, given that event \(S_{1}\) or \(S\), has occurred are $$ P\left(A \mid S_{1}\right)=.2 \text { and } P\left(A \mid S_{2}\right)=.3 $$ Use this information to answer the questions in Exercises \(1-3 .\) Use Bayes' Rule to find \(P\left(S_{2} \mid A\right)\).

Short answer

Answer: The probability is approximately 0.3913 or 39.13%.

Step-by-step solution

Understand Bayes' Rule

Bayes' Rule is a method to update the probability of a hypothesis based on new evidence. In this context, it helps us to find the probability of \(S_2\) occurring given that event A has occurred. Mathematically, Bayes' Rule is defined as: $$ P(S_2 | A) = \frac{P(A | S_2) * P(S_2)}{P(A)} $$ Here, we are given the \(P(A | S_1)\), \(P(A | S_2)\) and \(P(S_1)\), and \(P(S_2)\) values, but we need to find the value of \(P(A)\) before we can apply Bayes' Rule.

Calculate P(A) using the Law of Total Probability

The Law of Total Probability states that if we have a partition of the sample space (in this case, the partition is \(S_1\) and \(S_2\)), then P(A) can be calculated using the following formula: $$ P(A) = P(A | S_1)*P(S_1) + P(A | S_2)*P(S_2) $$ We can plug in the given values to find P(A): $$ P(A) = 0.2*0.7 + 0.3*0.3 $$ $$ P(A) = 0.14 + 0.09 = 0.23 $$

Apply Bayes' Rule

Now, we have all the necessary values to apply Bayes' Rule: $$ P(S_2 | A) = \frac{P(A | S_2) * P(S_2)}{P(A)} $$ Plug in the values and calculate the result: $$ P(S_2 | A) = \frac{0.3*0.3}{0.23} = \frac{0.09}{0.23} $$ $$ P(S_2 | A) \approx 0.3913 $$

Interpret the Result

The probability of selecting a sample from population \(S_2\), given that event A has occurred, is approximately 0.3913 or 39.13%.

Key concepts

Probability Rules

Probability rules are the foundational principles of probability that help us calculate and understand various probabilistic events. These rules are essential for solving problems in probability, and they are frequently used in conjunction with other concepts like Bayes' Theorem.

Some key probability rules include:

• Addition Rule: Useful when determining the probability of either of two mutually exclusive events occurring. If events A and B cannot happen at the same time, then the probability of A or B is simply the sum of their probabilities: \( P(A \cup B) = P(A) + P(B) \).
• Multiplication Rule: When the events are independent, the probability of both events A and B occurring is the product of their respective probabilities: \( P(A \cap B) = P(A) \cdot P(B) \).
• Complement Rule: The probability of an event not occurring is equal to one minus the probability of the event occurring: \( P(A') = 1 - P(A) \).

Understanding and using these basic rules correctly is crucial for solving more complex problems involving events and probabilities.

Law of Total Probability

The Law of Total Probability is a vital tool in probability theory, particularly when dealing with conditional probabilities. It allows us to calculate the overall probability of an event based on conditional probabilities given different initial conditions or events.

Imagine you have a sample space divided into different parts or "events," such as the populations \( S_1 \) and \( S_2 \) in our example. The Law of Total Probability can be expressed as:

• \( P(A) = P(A | S_1) \cdot P(S_1) + P(A | S_2) \cdot P(S_2) \)

To apply this law, follow these steps:

• Identify and list all possible partitions of the sample space that the event can occur.
• Calculate the probability of the event for each partition, multiplying by the probability of the partition itself.
• Add up these partial probabilities to arrive at the total probability of the event.

In our case, we found the probability of the event \( A \) occurring, which provides us the groundwork to apply Bayes' Theorem effectively.

Conditional Probability

Conditional probability refers to the probability of an event occurring given that another event has already occurred. It forms the basis for concepts like Bayes' Theorem and is denoted as \( P(A | B) \), which reads as "the probability of A given B."

Conditional probability is crucial for assessing the likelihood of an event when the outcome is influenced by prior events. The formula for conditional probability is generally given by:

• \( P(A | B) = \frac{P(A \cap B)}{P(B)} \)

In the context of our exercise, we are working to determine probabilities like \( P(S_2 | A) \), meaning "given that event \( A \) has already occurred, what is the probability of \( S_2 \)?"

Here’s how you can approach conditional probability problems:

• Understand the dependencies between events.
• Use the given data to calculate probabilities of intersection and the known event.
• Apply the formula to determine the conditional probability.

In practice, conditional probability helps in making informed decisions where part of the outcome is already known, refining predictions in a variety of fields.