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

Short answer

Answer: The probability of event A occurring is \(0.23\).

Step-by-step solution

Identify the given probabilities

We are given that: - \(P(S_1) = 0.7\) - \(P(S_2) = 0.3\) - \(P(A \mid S_1) = 0.2\) - \(P(A \mid S_2) = 0.3\)

Use the Law of Total Probability

We can now apply the Law of Total Probability formula: $$P(A) = P(A \mid S_1)P(S_1) + P(A \mid S_2)P(S_2)$$

Plug in the values

Substitute the given values into the formula: $$P(A) = (0.2)(0.7) + (0.3)(0.3)$$

Calculate P(A)

Perform the calculations: $$P(A) = 0.14 + 0.09$$ $$P(A) = 0.23$$ So the probability that event A occurs, \(P(A)\), is \(0.23\).

Key concepts

Conditional Probability

When studying probability, a fundamental concept is known as conditional probability. It refers to the probability of an event occurring, given that another event has already occurred. This is notated as P(A | B), which reads as 'the probability of A given B'.

For example, if we have two populations, S1 and S2, and we're interested in the occurrence of an event A, we need to consider the chances of A happening within each separate population. Conditional probability allows us to update our predictions about A according to whether we know the sample came from S1 or S2. In our exercise, we have the conditional probabilities as P(A | S1) and P(A | S2), suggesting different likelihoods of A depending on the source population.

The concept is particularly useful when dealing with complex scenarios where the probability of an outcome can be influenced by various factors, here embodied by the different source populations. Understanding conditional probability is essential for accurately calculating the chances of an event under certain conditions, a skill not just for textbook exercises but in real-world statistical analysis as well.

Probability of an Event

The probability of an event is a measure of the likelihood that the event will occur. It is a value between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The probability of an event A is notated as P(A).

In our exercise, determining P(A) involves combining the probabilities of event A occurring in two separate populations. By applying the Law of Total Probability, we effectively weigh each conditional probability by the probability of its conditioning event, thus acknowledging each population's contribution to the overall event probability.

Diagramming the Probability

Think of your overall probability as a pie made up of slices that represent different contributing factors. In our case, the slices for event A are based on which population the sample comes from – with S1 or S2. Calculating P(A) with the Law of Total Probability is akin to piecing together the complete pie to understand the full picture of how likely event A is to occur.

Statistical Populations

A statistical population is a set of entities about which statistical inferences are to be made, often based on a sample from the population. In probability and statistics, a population doesn't necessarily refer to people; it could be a collection of objects, events, processes, measurements, or even the results of a model.

In our exercise, we deal with two different populations, S1 and S2. Each population may behave differently and thus has distinct probabilities associated with event A, P(A | S1) and P(A | S2) respectively. By identifying and working with these populations separately, we respect the different mechanisms at play within each group.

Importance in Analysis

Recognizing the diversity among populations is crucial for accurate data analysis. When calculating overall probabilities, failing to account for the unique characteristics of separate populations would lead to misleading conclusions. Our exercise showcases how to consider these differences systematically to arrive at a true understanding of the likelihood of event A.