Question

If \(A\) and \(B\) are independent and \(P(A)=1 / 2, P(B)=1 / 3\) then \(P(A \cap B)=\).

Short answer

The probability \(P(A \cap B)\) is \(\frac{1}{6}\).

Step-by-step solution

Understand Independence

Two events, \(A\) and \(B\), are said to be independent if the occurrence of one does not affect the probability of the occurrence of the other. In mathematical terms, for independent events, \(P(A \cap B) = P(A) \times P(B)\).

Identify Given Probabilities

We are given that \(P(A) = \frac{1}{2}\) and \(P(B) = \frac{1}{3}\). We will use these probabilities in our calculation.

Apply Independence Formula

Since \(A\) and \(B\) are independent, we calculate the probability of their intersection using the formula for independent events: \(P(A \cap B) = P(A) \times P(B)\).

Calculate Intersection Probability

Substitute the given probabilities into the formula: \(P(A \cap B) = \frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\).

Verify the Calculation

Re-evaluate the multiplication to ensure accuracy: \(\frac{1}{2} \times \frac{1}{3} = \frac{1}{6}\), confirming the result that has been computed is correct.

Key concepts

Independent Events

When we talk about independent events in probability, think of two dice being rolled. If you toss one die, the outcome does not impact the result of the other die. Now, let's put this idea into probabilities: two events, say \(A\) and \(B\), are independent if knowing that one occurred does not change the likelihood of the other occurring.
The mathematical definition is simple yet powerful: for independent events, the probability of both occurring is the product of their separate probabilities. That is:

• The formula for independent events: \(P(A \cap B) = P(A) \times P(B)\)

This means if you know the probabilities of \(A\) and \(B\), you can easily find out the probability of both happening together without further information. This simplifies many calculations where independence can be assumed.

Intersection of Events

The intersection of events in probability is all about finding the chance of two events happening at the same time. Imagine two overlapping circles in a Venn diagram, where the overlapping part represents the intersection of two sets or events, \(A\) and \(B\).
So, in probability terms, \(P(A \cap B)\) is the probability that both \(A\) and \(B\) happen. For independent events, as seen in our exercise, this is straightforward to calculate because we leverage their independence.
The key point is:

• For independent events: \(P(A \cap B) = P(A) \times P(B)\)

The intersection operation is especially useful in situations involving multiple events where outcomes overlap. Understanding this concept is crucial when working with probabilities across disciplines.

Probability Calculation

Calculating probabilities involves some basic arithmetic once you know the types of events you're dealing with. In our example, we have been given probabilities for two independent events. We simply multiply them to find the probability of their intersection.
Let's walk through this:

• We know: \(P(A) = \frac{1}{2}\) and \(P(B) = \frac{1}{3}\).
• To find \(P(A \cap B)\), multiply these values: \( \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} \).

These calculations ensure that we directly find how common outcomes occur together based on given probabilities. Multiplication provides a simple and efficient way to compute this when dealing with independent events, saving time and effort in more complex situations.