# 9.8: Probability of Compound Events

**At Grade**Created by: CK-12

We begin this lesson with a reminder of probability.

The **experimental probability** is the ratio of the proposed outcome to the number of experiment trials.

\begin{align*}P(success)= \frac{number \ of \ times \ the \ event \ occured}{total \ number \ of \ trials \ of \ experiment}\end{align*}

Probability can be expressed as a percentage, a fraction, a decimal, or a ratio.

This lesson will focus on **compound events** and the formulas used to determine the probability of such events.

**Compound events** are two simple events taken together, usually expressed as \begin{align*}A\end{align*}

## Independent and Dependent Events

Example: Suppose you flip a coin and roll a die at the same time. What is the probability you will flip a head ** and** roll a four?

These events are **independent**. Independent events occur when the outcome of one event does not affect the outcome of the second event. Rolling a four has no effect on tossing a head.

To find the probability of two **independent events**, multiply the probability of the first event by the probability of the second event.

\begin{align*}P(A \ and \ B)=P(A) \cdot P(B)\end{align*}

Solution:

\begin{align*}P(tossing \ a \ head)&=\frac{1}{2}\\ P(rolling \ a \ 4)&=\frac{1}{6}\\ P(tossing \ a \ head \ AND \ rolling \ a \ 4)&=\frac{1}{2} \times \frac{1}{6}=\frac{1}{12}\end{align*}

When events depend upon each other, they are called **dependent events**. Suppose you randomly draw a card from a standard deck then randomly draw a second card **without replacing the first**. The second probability is now different from the first.

To find the probability of two **dependent events**, multiply the probability of the first event by the probability of the second event, *after the first event occurs*.

\begin{align*}P(A \ and \ B)=P(A) \cdot P(B \ following \ A)\end{align*}

Example: Two cards are chosen from a deck of cards. What is the probability that they both will be face cards?

Solution: Let \begin{align*}A = 1st \ Face \ card \ chosen\end{align*}

\begin{align*}P(A)&= \frac{12}{52}\\ P(B) & = \frac{11}{51}, \ \text{remember, one card has been removed.}\end{align*}

\begin{align*}P(A \ AND \ B)= \frac{12}{52} \times \frac{11}{51} \ & or \ P(A \cap B) = \frac{12}{52} \times \frac{11}{51} =\frac{33}{663}\\ P(A \cap B) & = \frac{11}{221}\end{align*}

## Mutually Exclusive Events

Events that **cannot happen at the same time** are called **mutually exclusive** events. For example, a number cannot be both even and odd or you cannot have picked a single card from a deck of cards that is both a ten and a jack. **Mutually inclusive** events, however, can occur at the same time. For example a number can be both less than 5 and even or you can pick a card from a deck of cards that can be a club and a ten.

When finding the probability of events occurring at the same time, there is a concept known as the “double counting” feature. It happens when the intersection is counted twice.

In mutually exclusive events, \begin{align*}P(A \cap B)=\phi\end{align*}

To find the probability of either mutually exclusive events \begin{align*}A\end{align*}

To find the probability of one or the other mutually exclusive or inclusive events, add the individual probabilities and subtract the probability they occur at the same time.

\begin{align*}P(A \ or \ B)=P(A)+P(B)-P(A \cap B)\end{align*}

Example: Two cards are drawn from a deck of cards. Let:

\begin{align*}A\end{align*}

\begin{align*}B\end{align*}

\begin{align*}C\end{align*}

Find the following probabilities:

(a) \begin{align*}P(A \ \text{or} \ B)\end{align*}

(b) \begin{align*}P(B \ \text{or} \ A)\end{align*}

(c) \begin{align*}P(A \ \text{and} \ C)\end{align*}

Solution:

(a) \begin{align*}P(A \ or \ B)=\frac{13}{52}+\frac{4}{52}-\frac{1}{52}\!\\
P(A \ or \ B) =\frac{16}{52}\!\\
P(A \ or \ B) =\frac{4}{13}\end{align*}

(b) \begin{align*}P(B \ or \ A)= \frac{4}{52} + \frac{13}{52}-\frac{1}{52}\!\\
P(B \ or \ A) = \frac{16}{52}\!\\
P(B \ or \ A) = \frac{4}{13}\end{align*}

(c) \begin{align*}P(A \ and \ C) = \frac{13}{52} \times \frac{13}{52}\!\\ P(A \ and \ C) = \frac{169}{2704}\!\\ P(A \ and \ C) = \frac{1}{16}\end{align*}

## Practice Set

- Define
*independent events*.

Are the following events independent or dependent?

- Rolling a die and spinning a spinner
- Choosing a book from the shelf then choosing another book without replacing the first
- Tossing a coin six times then tossing it again
- Choosing a card from a deck, replacing it, and choosing another card
- If a die is tossed twice, what is the probability of rolling a 4 followed by a 5?
- Define
*mutually exclusive*.

Are these events mutually exclusive or mutually inclusive?

- Rolling an even and an odd number on one die.
- Rolling an even number and a multiple of three on one die.
- Randomly drawing one card and the result is a jack and a heart.
- Randomly drawing one card and the result is black and a diamond.
- Choosing an orange and a fruit from the basket.
- Choosing a vowel and a consonant from a Scrabble bag.
- Two cards are drawn from a deck of cards. Determine the probability of each of the following events:
- \begin{align*}P\end{align*}(heart or club)
- \begin{align*}P\end{align*}(heart and club)
- \begin{align*}P\end{align*}(red or heart)
- \begin{align*}P\end{align*}(jack or heart)
- \begin{align*}P\end{align*}(red or ten)
- \begin{align*}P\end{align*}(red queen or black jack)

- A box contains 5 purple and 8 yellow marbles. What is the probability of successfully drawing, in order, a purple marble and then a yellow marble?
*{Hint: In order means they are not replaced.}* - A bag contains 4 yellow, 5 red, and 6 blue marbles. What is the probability of drawing, in order, 2 red, 1 blue, and 2 yellow marbles?
- A card is chosen at random. What is the probability that the card is black and is a 7?

**Mixed Review**

- A circle is inscribed within a square, meaning the circle's diameter is equal to the square’s side length. The length of the square is 16 centimeters. Suppose you randomly threw a dart at the figure. What is the probability the dart will land in the square, but not in the circle?
- Why is \begin{align*}7-14x^4+7xy^5-1x^{-1}=8x^2 y^3\end{align*} not considered a polynomial?
- Factor \begin{align*}72b^5 m^3 w^9-6(bm)^2 w^6\end{align*}.
- Simplify \begin{align*}2^5-7^3 a^3 b^7+3^5 a^3 b^7-2^3\end{align*}.
- Bleach breaks down cotton at a rate of 0.125% with each application. A shirt is 100% cotton.
- Write the equation to represent the percentage of cotton remaining after \begin{align*}w\end{align*} washes.
- What percentage remains after 11 washes?
- After how many washes will 75% be remaining?

- Evaluate \begin{align*}\frac{(100 \div 4 \times 2-49)^2}{9-2 \times 3+2^2}\end{align*}.