Question

Suppose \(P(A)=.1\) and \(P(B)=.5 .\) $$\text { If } P(A \mid B)=.1, \text { what is } P(A \cap B) ?$$

Short answer

Answer: The probability of the intersection of events A and B is P(A ∩ B) = 0.05.

Step-by-step solution

Identify given probabilities

We are given the following probabilities: $$P(A) = 0.1$$ $$P(B) = 0.5$$ $$P(A \mid B) = 0.1$$

Use the formula for conditional probability

We are given the conditional probability \(P(A \mid B)\), and we want to find \(P(A \cap B)\). We can use the formula for conditional probability to do this: $$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$$

Solve for P(A ∩ B)

Plug in the given values for \(P(A \mid B)\) and \(P(B)\) into the conditional probability formula and solve for \(P(A \cap B)\): $$0.1 = \frac{P(A \cap B)}{0.5}$$ To find \(P(A \cap B)\), multiply both sides of the equation by \(P(B) = 0.5\): $$P(A \cap B) = 0.1 \times 0.5 = 0.05$$

State the final answer

The probability of the intersection of A and B is: $$P(A \cap B) = 0.05$$

Key concepts

Intersection of Events

In probability theory, the intersection of events is an important concept. It refers to the probability that two (or more) events happen at the same time. If we have two events, A and B, the intersection, denoted as \(P(A \cap B)\), represents the probability that both events A and B occur together.Think of the intersection as the overlap between two events. If you imagine sets, the intersection is the part where both sets meet. In terms of our exercise, calculating the intersection of events tells us the likelihood of both events happening concurrently.- This concept is crucial when dealing with multiple probabilities and figuring out how they influence each other simultaneously.

Probability Formula

Mathematicians use several important probability formulas to calculate the chances of events occurring. One key formula is for calculating the probability of the intersection of two events, using conditional probability.In our exercise, we used the conditional probability formula:\[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]This equation tells us the conditional probability \(P(A \mid B)\), which is the probability of event A occurring, given that B has already happened. By reorganizing this formula, we can solve for \(P(A \cap B)\):- Multiply both sides by \(P(B)\) to isolate \(P(A \cap B)\) on one side. This allows us to find the exact probability of the intersection of events.

Conditional Probability Calculation

Conditional probability calculation helps us determine how one event's occurrence affects the likelihood of another event. It is crucial when particular conditions influence outcomes.To calculate conditional probability, you use the formula: \[P(A \mid B) = \frac{P(A \cap B)}{P(B)}\]Here, \(P(A \mid B)\) represents the probability of A occurring given that B has already occurred. The values for \(P(A \mid B)\) and \(P(B)\) can be plugged into this formula, allowing us to solve for \(P(A \cap B)\).

• Start by recognizing known values, such as \(P(A \mid B)\) and \(P(B)\).
• Substitute these values into the formula to find \(P(A \cap B)\).
• Perform basic arithmetic operations to isolate \(P(A \cap B)\) and determine its value.

This step-by-step approach helps break down complex scenarios into smaller, manageable calculations.