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 \(P\left(S_{1} \mid A\right)\).

Short answer

Question: Using Bayes' Rule, find the probability of event \(S_{1}\) occurring given that event A has occurred, given the following probabilities: \(P(S_{1}) = 0.7\), \(P(S_{2}) = 0.3\), \(P(A|S_{1}) = 0.2\), and \(P(A|S_{2}) = 0.3\). Answer: The probability of event \(S_{1}\) occurring given that event A has occurred is approximately \(0.6087\).

Step-by-step solution

Recall Bayes' Rule

Bayes' Rule is given by the following formula: $$ P(S_{1}|A) = \frac{P(A|S_{1})P(S_{1})}{P(A)} $$ Note that we have \(P(A|S_{1})\) and \(P(S_{1})\) given in the problem statement. We first need to find \(P(A)\), which can be calculated using the Law of Total Probability.

Use the Law of Total Probability to find \(P(A)\)

The Law of Total Probability states that: $$ P(A) = P(A|S_{1})P(S_{1}) + P(A|S_{2})P(S_{2}) $$ We are given all the probabilities in this formula, so we can plug them in and find \(P(A)\): $$ P(A) = (0.2)(0.7) + (0.3)(0.3) $$

Calculate \(P(A)\)

Compute the value of \(P(A)\): $$ P(A) = (0.2)(0.7) + (0.3)(0.3) = 0.14 + 0.09 = 0.23 $$

Apply Bayes' Rule formula to find \(P(S_{1}|A)\)

Now that we have all the probabilities, we can plug them back into the Bayes' Rule formula: $$ P(S_{1}|A) = \frac{P(A|S_{1})P(S_{1})}{P(A)} = \frac{(0.2)(0.7)}{0.23} $$

Calculate \(P(S_{1}|A)\)

Compute the value of \(P(S_{1}|A)\): $$ P(S_{1}|A) = \frac{(0.2)(0.7)}{0.23} = \frac{0.14}{0.23} \approx 0.6087 $$ So, the probability of event \(S_{1}\) occurring given that event A has occurred is approximately \(0.6087\).

Key concepts

Conditional Probability

Conditional probability is a fundamental concept in probability theory that deals with the probability of an event occurring given that another event has already occurred. To put it simply, it's the probability of event B happening when we already know that event A has taken place. Mathematically, it's expressed as
\( P(B|A) = \frac{P(A \cap B)}{P(A)} \),
where \( P(A \cap B) \) is the probability of both events A and B occurring, and \( P(A) \) is the probability of event A occurring.

In the context of our textbook exercise, \( P(A|S_1) \) would represent the probability of event A happening given that the sample is from population \( S_1 \). Understanding this concept is crucial for many statistical applications, including Bayes' Rule, which is used to update the probability of an event as more information becomes available.

Law of Total Probability

The law of total probability is an important rule in probability theory that allows us to calculate the probability of an event based on several mutually exclusive scenarios. It can be thought of as a way to 'break down' a probability across several distinct possibilities that cover all possible outcomes.
The formula for the law of total probability is:
\( P(A) = \sum P(A|B_i) \cdot P(B_i) \),
where \( B_1, B_2, ..., B_n \) represent the different exclusive scenarios, and \( P(A|B_i) \) is the conditional probability of A given \( B_i \).

In our exercise, we used the law of total probability to determine \( P(A) \), the probability of event A by taking into account both populations, \( S_1 \) and \( S_2 \). This step was essential for the subsequent application of Bayes' Rule.

Probability Theory

Probability theory is the branch of mathematics concerned with analyzing random phenomena and modeling uncertainty. It provides the foundation for statistical inference as well as many methods used in data science. The key principles of probability theory, such as random variables, probability distributions, and expected values, come together to help us understand and predict outcomes in situations where the results are not deterministic.
In the exercise, probability theory underpins the entire process of determining \( P(S_1|A) \), from understanding the individual probabilities given for \( S_1 \) and \( S_2 \), to applying Bayes' Rule for the final calculation. By having a good grasp of probability theory, students can adapt these concepts to solve similar problems where it's necessary to evaluate the likelihood of various events based on given data.