# Probability of A and B for Independent Events

## (Combined Events Part 1)

In this section we learn how to calculate the probability of $$A$$ and $$B$$ occuring, that's when both events $$A$$ and $$B$$ happen at the same time and we learn the formula for independent events $$A$$ and $$B$$. Although we sometimes refer to this probability with the notation $$p\begin{pmatrix}A \ \text{and} \ B \end{pmatrix}$$ we'll use the correct mathematical notation and write: $p\begin{pmatrix}A \cap B \end{pmatrix}$ To be clear, $$p\begin{pmatrix}A \cap B \end{pmatrix}$$ should be read "the probability of A and B occurring".

#### Venn Diagram for $$p\begin{pmatrix}A \cap B \end{pmatrix}$$

The Venn diagram shown here shows two sets $$A$$ and $$B$$, which correspond to two events $$A$$ and $$B$$. The two sets overlap, we say they intersect, thereby creating the set $$A\cap B$$ (that's the event "$$A$$ and $$B$$").

If we're given, or can find, how many elements are inside the set $$A \cap B$$ and we know how many elements are inside the universal set $$U$$, that's $$n\begin{pmatrix}U \end{pmatrix}$$, then we can calculate $p\begin{pmatrix}A \cap B \end{pmatrix} = \frac{n \begin{pmatrix} A \cap B \end{pmatrix}}{n \begin{pmatrix}U\end{pmatrix}}$ We frequently won't be given enough information to find both $$n\begin{pmatrix}A \cap B \end{pmatrix}$$ and $$n\begin{pmatrix} U \end{pmatrix}$$. Instead we'll "just" be given, or will be able to find the probabilities $$p\begin{pmatrix}A\end{pmatrix}$$ and $$p\begin{pmatrix}B \end{pmatrix}$$. In such cases we'll have to use the formula that we learn next.

## Formula for $$p\begin{pmatrix} A \cap B\end{pmatrix}$$ - Independent Events

The formula we learn here is for independent events. If you're not sure what independent events are then do make sure to click on the "Learn More" button below.

Given two events $$A$$ and $$B$$ that are independent, then provided we are given (or can find) the probabilities of both events, $$p\begin{pmatrix}A \end{pmatrix}$$ and $$p\begin{pmatrix}B \end{pmatrix}$$, we can calculate the probability $$p\begin{pmatrix}A\cap B \end{pmatrix}$$ using the following formula

### Formula - $$p\begin{pmatrix}A \cap B \end{pmatrix}$$ for Independent Events

Given two independent events $$A$$ and $$B$$, the probability of the event "$$A$$ and $$B$$" occurring is written $$p\begin{pmatrix}A \cap B \end{pmatrix}$$ and is given by the formula: $p\begin{pmatrix}A \cap B \end{pmatrix} = p\begin{pmatrix}A \end{pmatrix} \times p\begin{pmatrix}B \end{pmatrix}$

### Tutorial: Multiplication Rule for $$p\begin{pmatrix}A \cap B \end{pmatrix}$$

The following tutorial we see, through a couple of examples, how to use the multplcation rule for independent events.

### Examples

1. Question: Two unbiased dice, both numbered $$1$$ to $$6$$, are rolled. What is the probability of rolling two $$4$$'s?

Answer: The events "rolling a 4 with the first die" and "rolling a 4 with the second die" are independent. Indeed, whatever the outcome of one die has no impact on the likelihood of rolling a 4 with the second die. We can therefore use the formula stated above to write: \begin{aligned} p\begin{pmatrix}\text{rolling two 4's} \end{pmatrix} & = p\begin{pmatrix} \text{rolling a 4 with one die} \end{pmatrix} \times p\begin{pmatrix} \text{rolling a 4 with the other die} \end{pmatrix} \\ & = \frac{1}{6} \times \frac{1}{6} \\ p\begin{pmatrix}\text{rolling two 4's} \end{pmatrix} & = \frac{1}{36} \end{aligned}

2. Question: A bag contains $$3$$ orange marbles and $$4$$ green marbles. A marble is picked at random and, once its color is noted, put bag in the bag. A second marble is then picked at random and its color noted as well. What is the probability of picking two orange marbles?
Answer: Since the first marble that is picked is put back in the back, whatever color marble is picked first will have no impact on what is picked second. The events "1st marble is orange" and "2nd marble is orange" are therefore independent and we can define those two events as:
• $$A$$: picking an orange on the first draw.
• $$B$$: picking an orange on the second draw.
Given there are $$3$$ orange marbles and $$7$$ marbles in total, we quickly see that $$p\begin{pmatrix}A\end{pmatrix} = \frac{3}{7}$$ and $$p\begin{pmatrix}B\end{pmatrix} = \frac{3}{7}$$. Finally, using the formula we saw further-up we can therefore calculate the probability of picking two orange marbles: \begin{aligned} p\begin{pmatrix}A \cap B \end{pmatrix} & = p\begin{pmatrix}A\end{pmatrix}\times p\begin{pmatrix}A\end{pmatrix} \\ & = \frac{3}{7} \times \frac{3}{7} \\ & = \frac{3\times 3}{7\times 7} \\ p\begin{pmatrix}A \cap B \end{pmatrix} & = \frac{9}{49} \end{aligned}

## Exercise 1

1. For a survey, a group of people, with pets at home, was asked whether they had a cat or a dog. The result is summarized in the Venn diagram, shown here.

1. How many people participated in the survey?
2. How many participants own a cat?
3. How many people own both a cat and a dog?
4. What is the probability that one of the people, participating in the survey, owns both a cat and a dog?

2. The probability that Helen do well at her Mathematics test is $$p\begin{pmatrix}M\end{pmatrix} = 0.8$$. The probability that she do well at her French test is $$p\begin{pmatrix}F\end{pmatrix} = 0.7$$.
1. What is the probability that Helen does well at both her Mathematics test and her French test?
2. What is the probability that Helen doesn't do well at her Mathematics test and does well at her French test?

3. Like every Sunday afternoon, Cathy and Jonathan play a game of darts. On any given turn, the probability that Cathy hits the center of the target is $$0.5$$ and the probability that they both hit the center of the target is $$0.2$$. What is the probability that Jonathan hit the center of the target?

4. A fair coin is tossed and an unbiased die is rolled. Find the probability of:
1. rolling an even number.
2. flipping tails.
3. rolling an even number and flipping tails.

Note: this exercise can be downloaded as a worksheet to practice with: