Question

NUMBER SENSE Events \(A\) and \(B\) are dependent. Suppose \(P(B \mid A)=0.6\) and \(P(A\) and \(B)=0.15\). Find \(P(A)\).

Short answer

The probability of event A, P(A), is 0.25.

Step-by-step solution

Understand the Problem

In the problem, it is given that \(P(B \mid A) = 0.6\) and \(P(A \text{ and } B) = 0.15\). These are instances of dependent events in probability, meaning that the occurrence of event A has an effect on the likelihood of event B. Our task is to find \(P(A)\). To do this, the formula for conditional probability needs to be rearranged by multiplying both sides by \(P(A)\) to isolate \(P(A)\) on one side of the equation.

Rearrange the Formula for Conditional Probability

The formula for conditional probability is \(P(B \mid A) = \frac{P(A \text{ and } B)}{P(A)}\). By multiplying both sides by \(P(A)\), we obtain: \(P(A) \cdot P(B \mid A) = P(A \text{ and } B)\). With the numbers given in the exercise, this equation is: \(P(A) \cdot 0.6 = 0.15\).

Solve for \(P(A)\)

The last step is to solve for \(P(A)\). This can be done by dividing both sides by 0.6. This gives: \(P(A) = \frac{0.15}{0.6} = 0.25\)

Key concepts

Dependent Events in Probability

When studying probability, understanding the concept of dependent events is crucial. These are events where the occurrence of one event affects the probability of the occurrence of another. For instance, if you pull a card from a deck of cards, the probability of the second card you pull being an ace depends on whether the first card was an ace or not.

In mathematical terms, we denote the probability of event B occurring given that event A has already occurred as \( P(B \mid A) \). The expression “being dependent” means that \( P(B \mid A) \) is not equal to \( P(B) \), the latter representing the probability of B occurring without any prior event.

This relationship between events is central to problems involving conditional probability, where we compute probabilities taking into account certain conditions or previous events. In the given exercise, knowing that \( P(A) \) influences \( P(B \mid A) \), and vice versa, is key to finding the solution.

Probability of an Event

The probability of an event is a measure of how likely that event is to occur. In a formal probabilistic framework, this is quantified as a number between 0 and 1, where 0 signifies an impossible event and 1 signifies a certain event. For example, the probability of rolling a 4 on a fair six-sided die is \( \frac{1}{6} \), since there is one 4 on the die and six possible outcomes.

To calculate the probability of an event, one generally counts the number of ways that event can occur and divides it by the total number of possible outcomes. Going back to our exercise, we were asked to find \( P(A) \), where \( P(A) \) is the probability of event A occurring. The key to solving the problem was understanding how the probability of A could be extracted from the joint probability of A and B (dependent events) using the definitions and properties of conditional probability.

Mathematical Reasoning

Applying mathematical reasoning requires a logical thought process to solve problems using mathematical concepts and techniques. It involves understanding relationships, constructing logical arguments, and arriving at conclusions based on given information. In probability, this often means manipulating equations to isolate the desired variable, just as we did in solving our textbook exercise.

When we encounter a problem, such as finding \( P(A) \) given \( P(B \mid A) \) and \( P(A \text{ and } B) \), we have to work backwards, deducing from what we know (the conditional probability and the joint probability) to what we need to find out (the probability of A alone). Through mathematical reasoning, we recognized that rearranging the formula for conditional probability would allow us to solve for \( P(A) \), translating our conceptual understanding into a numerical answer. This skill is fundamental, not just in probability, but in all areas of mathematics.