#### Objective

In this lesson, you will learn about calculating the probability that any one of multiple ** mutually exclusive independent events** will occur in a single experiment.

#### Concept

If you think it through, it should make sense that the probability of pulling one Queen at random from a standard deck is

After this lesson on the *union of compound events*, we’ll return to this question and work out the answer.

#### Watch This

http://youtu.be/QE2uR6Z-NcU Khan Academy – Addition Rule for Probability

#### Guidance

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

the probability of*Intersection:**both*or*all*of the events you are calculating happening at the same time (less likely).the probability of*Union:**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:

This can also be thought of as:

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.

**Example A**

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?

**Solution:** Recall the formula for the union of simple probabilities:

In this case, we have:

Which would reduce to:

**Example B**

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

**Solution:** 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:

Reducing to:

**Example C**

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?

**Solution:** 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:

**Concept Problem Revisited**

*It should make sense now that the probability of pulling one Queen at random from a standard deck is 452 or 113, 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:

#### Vocabulary

A ** statistical probability** is the calculated probability that a given outcome will occur if the same experiment were completed an infinite number of times.

The probability of the ** union** of multiple, mutually exclusive, events is the sum of the probabilities of each of the individual outcomes occurring during a given trial.

An ** event** is any collection of the outcomes of an experiment.

** Mutually exclusive** events cannot occur at the same time (they have no overlap). For instance, a single coin flip cannot be

*both*heads and tails.

An ** independent event** is an event that is unaffected by any other event occurring before or after it.

An ** outcome** is the result of a single trial.

#### Guided Practice

- 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?
- What is the probability of a spinner landing on “2”, “3”, or “6” if there are 6 equally spaced points on the spinner?
- What is the probability of pulling a red or black card at random from a standard deck?
- 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?
- 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

**Solutions:**

P(red or blue)=1 red marble+1 blue marble12 total marbles=212 or 16 or 16.6% P(2 or 3 or 6)=1 number 6+1 number 2+1 number 36 total numbers=36 or 12 or 50% P(red or black)=26 red cards+26 black cards52 total cards=5252=11 or 100% P(red or green)=5 red marbles+7 green marbles32 total marbles=1232 or 616 or 38 or 37.5% P(red)=6 females wearing red+9 males wearing red74 total students=1574 or 20.3%

#### Practice

- 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?

### Answers for Explore More Problems

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