Question

Use the information that, for events \(\mathrm{A}\) and \(\mathrm{B},\) we have \(P(A)=0.4, P(B)=0.3,\) and \(P(A\) and \(B)=0.1.\) Find \(P(A\) if \(B).\)

Short answer

The conditional probability \(P(A|B)\) is approximately 0.333

Step-by-step solution

Understand the symbols

This step includes understanding the symbols in the formula. \(P(A|B)\) represents the probability of event A given event B, \(P(A \cap B)\) represents the probability of both A and B occurring, and \(P(B)\) is the probability of B occurring.

Insert the given values into the formula

Here, you will use the provided values for the probabilities. So, \(P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.1}{0.3}\).

Simplify the fraction

Finally, reduce the fraction to its simplest form to get the final probability. So, \(P(A|B) = \frac{0.1}{0.3} = \frac{1}{3} = 0.333\).

Key concepts

Probability Theory

Probability theory is a fundamental aspect of mathematics that deals with the estimation of the likelihood of certain events. It's a framework that allows us to quantify the uncertainty of phenomena and make educated guesses about the outcomes of various situations. The basic concept includes understanding the probability of a single event, which is expressed as a number between 0 and 1, where 0 means the event will never happen, and 1 means it will always happen.

For example, flipping a fair coin has two possible outcomes: heads or tails. Since both outcomes are equally likely, each has a probability of 0.5 or 50%. In the context of the given exercise, you're working with probabilities assigned to events A and B. Analyzing how these events interact and the likelihood of their occurrence provides a practical application of probability theory. The calculation presented in the solution involves conditional probability, a concept that evaluates the probability of an event given that another event has occurred.

Bayes' Theorem

Bayes' theorem is a powerful tool in probability theory, named after the statistician Thomas Bayes. It's used to update the probability of an event as more information becomes available. This theorem incorporates prior knowledge or belief, along with new evidence, to arrive at a posterior probability.

The theorem is mathematically represented as:\[ P(A|B) = \frac{P(B|A) \times P(A)}{P(B)} \]
Here, \(P(A|B)\) is the probability of event A occurring given that B has occurred, \(P(B|A)\) is the probability of event B given event A, \(P(A)\) is the probability of event A, and \(P(B)\) is the probability of event B. This theorem is essential when the conditional probability is not directly observable, and we need to compute it using the inverse condition.In the context of the textbook solution, Bayes' theorem shapes how we think about updates to our understanding of event A happening when we have knowledge about event B.

Event Independence

Understanding event independence is crucial in probability theory. Two events, A and B, are considered independent if the occurrence of one does not affect the probability of the occurrence of the other. This can be mathematically expressed as follows:\[ P(A \text{ and } B) = P(A) \times P(B) \]
However, if events are not independent, this equation does not suffice. In such cases, the occurrence of one event does affect the likelihood of the other, which is particularly important when calculating conditional probabilities.

In our exercise, we were not explicitly told whether events A and B are independent or not. But the details given suggest a level of dependency because the probability of A and B occurring together (\(P(A \text{ and } B) = 0.1\)) is different from the product of their individual probabilities (\(P(A) \times P(B)\)). This is an essential consideration when applying probabilities in real-world scenarios, as it helps inform the prediction of complex event outcomes.