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# Additive and Multiplicative Rules for Probability

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Additive and Multiplicative Rules for Probability

Do you remember Alicia from the Use Tree Diagrams to List All Possible Outcomes Concept? Take a look at the tree diagram of outfits.

Alicia is going to sing for the Talent Show. She is very excited and has selected a wonderful song to sing. She has been practicing with her singing teacher for weeks and is feeling very confident about her ability to do a wonderful job.

Her performance outfit is another matter. Alicia has selected a few different skirts and a few different shirts and shoes to wear. Here are her options for shirts

Striped shirt

Solid shirt

Here are her options for skirts.

Blue skirt

Red skirt

Brown skirt

Here are her options for shoes

Dance shoes

Black dress shoes

Here is a tree diagram of Alicia's outfits.

What is the probability of Alicia wearing her striped shirt, blue skirt and dance shoes?

To figure this out you will need to know how to use a tree diagram to calculate a specific probability. Pay attention and you will learn what you need to know in this Concept.

### Guidance

Previously we worked on tree diagrams. We have seen how tree diagrams can be very helpful when looking for a sample space. Tree diagrams can also be helpful when finding probability.

Finding the probability of an event is a matter of finding the ratio of favorable outcomes to total outcomes. For example, the sample space for a single coin flip has two outcomes: heads and tails. So the probability of getting heads on any single coin flip is:

$P (\text{heads}) = \frac{favorable \ outcomes}{total \ outcomes} =\frac{1}{2}$

You can see that the sample space is represented by a number in the total outcomes. For example, if you had a spinner with four colors, the colors by name would be the sample space and the number four would be the total possible outcomes.

What about if we flipped a coin more than one time?

To find the probability of a single outcome for more than one coin flip, use a tree diagram to find all possible outcomes in the sample space.

Then count the number of favorable outcomes within that sample space to find the probability.

For example, to find the probability of tossing a single coin twice and getting heads both times, make a tree diagram to find all possible outcomes.

The diagram shows there are 8 total outcomes and they are paired with first toss option and second toss option.

Then pick out the favorable outcome–in this case, the outcome “heads-heads” is shown in red. You could have selected any of the favorable outcomes for the probability to be accurate.

Now write the ratio of favorable outcomes to total outcomes in the sample space.

$P (\text{heads-heads}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{4}$

You can see that since 1 of 4 outcomes is a favorable outcome, the probability of the coin landing on heads 2 times in a row is $\frac{1}{4}$ .

Let’s look at another scenario.

What is the probability of flipping a coin two times and getting two matching results–that is, either two heads or two tails?

First, let’s create a tree diagram to see our options.

Once again, just pick out the favorable outcomes on the same tree diagram. They are shown in red.

You can see that 2 of 4 total outcomes match.

$P (2 \ \text{heads or 2 tails}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{2}{4}=\frac{1}{2}$

You can see that the probability of flipping two heads or two tails is 1:2.

Try a few on your own.

#### Example A

Look at the tree diagram above. What is the probability of it being heads and then tails?

Solution: $\frac{1}{4}$

#### Example B

What is the probability of tails then heads?

Solution: $\frac{1}{4}$

#### Example C

What is Example A and B as a percent?

Solution: $25%$

Here is the original problem once again.

Alicia is going to sing for the Talent Show. She is very excited and has selected a wonderful song to sing. She has been practicing with her singing teacher for weeks and is feeling very confident about her ability to do a wonderful job.

Her performance outfit is another matter. Alicia has selected a few different skirts and a few different shirts and shoes to wear. Here are her options for shirts

Striped shirt

Solid shirt

Here are her options for skirts.

Blue skirt

Red skirt

Brown skirt

Here are her options for shoes

Dance shoes

Black dress shoes

Here is a tree diagram of Alicia's outfits.

What is the probability of Alicia wearing her striped shirt, blue skirt and dance shoes?

There are twelve possible outcomes, but only one is the striped shirt, blue skirt and dance shoes.

Our answer is $\frac{1}{12}$ .

### Vocabulary

Tree Diagram
a visual way of showing all of the possible outcomes of an experiment. Called a tree diagram because each option is drawn as a branch of a tree.
Sample Space
the possible outcomes in an experiment.
Favorable Outcome
the outcome that you are looking for in an experiment.
Total Outcome
the number of options in the sample space.

### Guided Practice

Here is one for you to try on your own.

What is the probability of a win-win-win?

There are eight possible outcomes for the teams.

There is one option for a win-win-win in all three games.

The probability is $\frac{1}{8}$ .

### Practice

Directions : Answer each question. Use tree diagrams when necessary.

1. What is the probability that the arrow of the spinner will land on red on a single spin?

2. If the spinner is spun two times in a row, what is the probability that the arrow will land on red both times?

3. If the spinner is spun two times in a row, what is the probability that the spinner will land on the same color twice?

4. If the spinner is spun two times in a row, what is the probability that the arrow will land on red at least one time?

5. If the spinner is spun two times in a row, what is the probability that the spinner will land on a different color both times?

6. If the spinner is spun two times in a row, what is the probability that the arrow will land on blue or green at least one time?

7. Two cards, the Ace and King of hearts, are taken from a deck, shuffled, and placed face down. What is the probability that a single card chosen at random will be an Ace?

8. If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards will be Kings?

9. If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards will match?

10. If one card is chosen from the 2-card stack above, then returned to the stack and a second card is chosen, what is the probability that both cards NOT match?

Directions : Look at the tree diagram and figure out each probability as a ratio.

11. What is the probability of the option tile, steel and granite?

12. What is the probability of either tile, steel, granite or tile, granite, white?

13. What is the probability of Formica being in the option?

14. What is the probability of tile and Formica being in the option?

15. What is the probability of white and Formica being in the option?

### Vocabulary Language: English

Favorable Outcome

Favorable Outcome

A favorable outcome is the outcome that you are looking for in an experiment.
Independent Events

Independent Events

Two events are independent if the occurrence of one event does not impact the probability of the other event.
multiplicative rule of probability

multiplicative rule of probability

The multiplicative rule of probability then states that P(AB) = P(B) x P(A/B)
Mutually Exclusive Events

Mutually Exclusive Events

Mutually exclusive events have no common outcomes.
Sample Space

Sample Space

In a probability experiment, the sample space is the set of all the possible outcomes of the experiment.
Total Outcomes

Total Outcomes

In probability, the total outcomes are the total number of possible outcomes for the probability experiment.
Tree Diagram

Tree Diagram

A tree diagram is a visual way of showing options and variables. The lines of a tree diagram look like branches on a tree.