If you think it through, it should make sense that the probability of pulling one Queen at random from a standard deck is \begin{align*}\frac{4}{52}\end{align*} or \begin{align*}\frac{1}{13}\end{align*}, since there are 4 Queens in a standard 52 card deck. How then would you calculate the probability of pulling a Queen OR a King from the same deck?

### Union of Compound Events

When multiple independent events may occur during a particular experiment, there are a couple of different types of outcomes you may need to consider:

**Intersection:**the probability of both or all of the events you are calculating happening at the same time (less likely).**Union:**the probability of any one of multiple events happening at a given time (more likely).

In this lesson, we will focus on **union**. Calculating the union is relatively easy, you just add up the individual probabilities of the events:

\begin{align*}P(x \ or \ y)=P(x)+P(y)\end{align*}

This can also be thought of as:

\begin{align*}P(x \ or \ y)=\frac{(\text{number of outcomes where} \ x \ \text{is true}) +(\text{number of outcomes where} \ y \ \text{is true})}{\text{total number of possible outcomes}}\end{align*}

It is really just that simple! It is intuitive also, assuming there is no overlap (which we will consider later), it just makes sense to think that if you have a 20% probability of one thing happening, and a 30% probability of another, then you have a 50% probability of one of the two of them happening during a given experiment.

**Calculating Probability **

1. You are given a big containing 15 equally sized marbles. You know there are 5 yellow marbles, 5 blue marbles, and 5 green marbles in the bag. What is the statistical probability that you would pull a yellow *or* green marble out, if you reach in the bag and grab a marble at random?

Recall the formula for the union of simple probabilities:

\begin{align*}P(x \ or \ y)=\frac{(\text{number of outcomes where} \ x \ \text{is true}) +(\text{number of outcomes where} \ y \ \text{is true})}{\text{total number of possible outcomes}}\end{align*}

In this case, we have:

\begin{align*}P(yellow \ or \ green)=\frac{5 \ yellow \ marbles+5 \ green \ marbles}{15 \ total \ marbles}\end{align*}

Which would reduce to:

\begin{align*}P(yellow \ or \ green)=\frac{2}{3} \ or \ 66.6 \bar{6} \%\end{align*}

2. What is the probability of rolling an odd or even number on a standard six-sided die?

A standard die has three odd numbers (1, 3, 5) and three even numbers (2, 4, 6). Therefore, the probability of rolling an odd or even number is:

\begin{align*}P(odd \ or \ even)=\frac{3 \ odd + 3 \ even}{6 \ total}=\frac{6}{6}\end{align*}

Reducing to:

\begin{align*}P(odd \ or \ even)=1 \ or \ 100 \%\end{align*}

3. If Lawrence is playing with a standard 52-card deck, what is the probability of pulling a 2, a 4, or a 6 out of the deck at random?

Let’s solve this one as the total of the individual probabilities. Lawrence’s probability of pulling a 2, 4, or 6 is the same as the union of the probability of each possible outcome:

\begin{align*}P(2, 4, \ or \ 6)=P(2)+P(4)+P(6)=\frac{1}{13}+\frac{1}{13}+\frac{1}{13}=\frac{3}{13} \ or \ 23.1 \%\end{align*}

**Earlier Problem Revisited**

*It should make sense now that the probability of pulling one Queen at random from a standard deck is \begin{align*}\frac{4}{52}\end{align*} or \begin{align*}\frac{1}{13}\end{align*}, since there are 4 Queens in a standard 52 card deck. How then would you calculate the probability of pulling a Queen OR a King from the same deck?*

Remember that the *union* of multiple probabilities is simple the total sum of all of the individual probabilities:

\begin{align*}P(Queen \ or \ King)=\frac{4 \ Queens+4 \ Kings}{52 \ Cards}=\frac{8}{52} \ or \ \frac{4}{26} \ or \ \frac{2}{13}=15.4 \%\end{align*}

### Examples

#### Example 1

What is the statistical probability of pulling either the only red or the only blue marble out of a bag with 12 marbles in it?

\begin{align*}P(red \ or \ blue)=\frac{1\text{ red marble} + 1\text{ blue marble}}{12\text{ total marbles}}=\frac{2}{12} \ or \ \frac{1}{6} \ or \ 16.6 \%\end{align*}

#### Example 2

What is the probability of a spinner landing on “2”, “3”, or “6” if there are 6 equally spaced points on the spinner?

\begin{align*}P(2 \ or \ 3 \ or \ 6)=\frac{1 \text{ number }6 + 1\text{ number }2 + 1\text{ number }3}{6\text{ total numbers}}=\frac{3}{6} \ or \ \frac{1}{2} \ or \ 50 \%\end{align*}

#### Example 3

What is the probability of pulling a red or black card at random from a standard deck?

\begin{align*}P(red \ or \ black)=\frac{26\text{ red cards} + 26\text{ black cards}}{52\text{ total cards}}=\frac{52}{52}=\frac{1}{1} \ or \ 100 \%\end{align*}

#### Example 4

What probability of picking a red or green marble from a bag with 5 red, 7 green, 6 blue, and 14 yellow marbles in it?

\begin{align*}P(red \ or \ green)=\frac{5\text{ red marbles}+7\text{ green marbles}}{32\text{ total marbles}}=\frac{12}{32} \ or \ \frac{6}{16} \ or \ \frac{3}{8} \ or \ 37.5 \%\end{align*}

#### Example 5

What is the of shaking the hand of a student wearing red if you randomly shake the hand of one person in a room containing the following mix of students?

- 13 female students wearing blue
- 7 male students wearing blue
- 6 female students wearing red
- 9 males students wearing red
- 18 female students wearing green
- 21 male students wearing green

\begin{align*}P(red)=\frac{6\text{ females wearing red}+9\text{ males wearing red}}{74\text{ total students}}=\frac{15}{74} \ or \ 20.3 \%\end{align*}

### Review

- What is the probability of rolling a standard die and getting between a 1 and 6 (inclusive)?
- What is the probability of pulling one card from a standard deck and it being an 8, a 3, or a queen?
- What is the probability of rolling a 5 or a 2 on an 8-sided die?
- What is the probability of pulling one card from a standard deck and it being a spade, a diamond, or a club?
- What is the probability of rolling a 1, 3, or 5 on a 7-sided die?
- What is the probability of pulling one card from a standard deck and it being a king, a 4, or a 8?
- What is the probability of pulling a yellow or blue candy from a bag containing 35 candies equally distributed among yellow, blue, green, red, and brown candies?
- What is the probability of spinning 2, 4, or 7 on a 10-space spinner (equally spaced)?
- What is the probability of rolling a 1, 3, 5, or 6 on a 20-sided die?
- A car factory creates cars in the following ratio: 3 green, 2 blue, 7 white, 2 black and 1 brown. What is the probability that a randomly selected car will be either blue or brown?
- There are 4 flavors of donuts on the shelf: glazed, sprinkles, plain, and powdered sugar. If there are equal numbers of each of the non-plain donuts, and half as many plain as any one of the others, what is the probability of randomly choosing a plain donut out of all donuts on the shelf?
- What is the probability of randomly choosing a red Ace or a black King from a standard deck?
- What is the probability of rolling a prime number or an even number on a standard die?
- Mr. Spence’s class has 13 students. 4 students are wearing coats, 3 are wearing vests, 3 are wearing hoodies, and the rest are in t-shirts. What is the probability that Mr. Spence will randomly call the name of a student wearing a coat or a vest?
- In the same class, what is the probability that Mr. Spence will randomly call the name of a student in a hoodie or t-shirt?

### Review (Answers)

To view the Review answers, open this PDF file and look for section 6.2.