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Additive and Multiplicative Rules for Probability

Apply tree diagrams to calculations.

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Additive and Multiplicative Rules for Probability
License: CC BY-NC 3.0

Abigail wants to pick out a bouquet of balloons for her best friend's birthday.  At the store, there is a jar with equal amounts of each of the following colors of balloons:  yellow, green, and blue. She randomly pulls and replaces balloons three times so that she can look at the colors.  What is the probability that Abigail will randomly choose one of each color over the three pulls?

In this concept, you will learn how to calculate the probability of one outcome from an event that occurs more than once.

Additive and Multiplicative Rules for 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:

\begin{align*}P (\text{heads}) = \frac{favorable \ outcomes}{total \ outcomes} =\frac{1}{2}\end{align*}

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 you 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, first 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.

Next 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.

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

\begin{align*}P (\text{heads-heads}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{4}\end{align*}

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 \begin{align*}\frac{1}{4}\end{align*}.  The answer is  \begin{align*}\frac{1}{4}\end{align*}.

Let’s look at another example.

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

First, create a tree diagram to see all options:

Next, 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.

Then, use the formula for probability of an event to calculate the probability:

\begin{align*}P (2 \ \text{heads or 2 tails}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{2}{4}=\frac{1}{2}\end{align*}

The answer is the probability of flipping two heads or two tails is 1:2.

Examples

Example 1

Earlier, you were given a problem about Abigail and her balloon bouquet.

She randomly pulls and replaces balloons three times from a jar that contains equal numbers of yellow, green, and blue.  What is the probability of her pulling one of each color after her last pull.

First, create a tree diagram of the possible outcomes:

1st Pull:                                                      yellow                                                      

2nd Pull:                        yellow                     green                        blue                                                 

3rd Pull:              yellow green blue        yellow green blue        yellow green blue

1st Pull:                                                       green

2nd Pull:                      yellow                       green                        blue

3rd Pull:               yellow green blue       yellow green blue        yellow green blue

1st Pull:                                                         blue

2nd Pull:                      yellow                        green                      blue

3rd Pull:               yellow green blue        yellow green blue       yellow green blue

Next, list all possible outcomes:

yellow-yellow-yellow, yellow-yellow-green, yellow-yellow-blue, yellow-green-yellow, yellow-green-green, yellow-green-blue, yellow-blue-yellow, yellow-blue-green, yellow-blue-blue

green-yellow-yellow, green-yellow-green, green-yellow-blue, green-green-yellow, green-green-green, green-green-blue, green-blue-yellow, green-blue-green, green-blue-blue

blue-yellow-yellow, blue-yellow-green, blue-yellow-blue, blue-green-yellow, blue-green-green, blue-green-blue, blue-blue-yellow, blue-blue-green, blue-blue-blue

Then, count the number of favorable outcomes (one of each color balloon) and total outcomes, and use the probability of favorable outcome formula to figure out the probability:

Favorable outcome = 6 times

Total outcomes = 27

\begin{align*}P ( \text{one of each color}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{6}{27}\end{align*}

The answer is the probability that Abigail will choose one of each color is \begin{align*}\frac{6}{27}\end{align*}.

Example 2

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

First, create a tree diagram of the possible outcomes:

Next, count the number of times the favorable outcome occurs:

Win-Win-Win occurs 1 time.

Then, using the formula for probability of favorable outcomes, calculate the probability:

\begin{align*}P ( \text{win-win-win}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{8}\end{align*} 

There are eight possible outcomes for the teams.

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

The answer is the probability of a win-win-win outcome is \begin{align*}\frac{1}{8}\end{align*}.

Use the diagram below to answer the questions that follow.


Example 3

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

First, locate the outcomes that are Heads-Tails.

Next, count the number of times this outcome occurs:

1 time.

Then, use the probability of favorable outcomes formula to calculate the probability:

\begin{align*}P (\text{heads-tails}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{4}\end{align*}

The answer is the probability of the outcome being heads and then tails is \begin{align*}\frac{1}{4}\end{align*}. 

Example 4

What is the probability of tails then heads?

First, locate the outcomes that are Tails-Heads.

Next, count the number of times this outcome occurs:

1 time

Then, use the probability of favorable outcomes formula to calculate the probability:

\begin{align*}P (\text{tails-heads}) = \frac{favorable \ outcomes}{total \ outcomes}=\frac{1}{4}\end{align*}

The answer is the probability of the outcome being tails and then heads is \begin{align*}\frac{1}{4}\end{align*}. 

Review

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?

  1. 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?
  2. 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?
  3. 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?
  4. 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?

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

  1. What is the probability of the option tile, steel and granite?
  2. What is the probability of either tile, steel, granite or tile, granite, white?
  3. What is the probability of Formica being in the option?
  4. What is the probability of tile and Formica being in the option?
  5. What is the probability of white and Formica being in the option?

Review (Answers)

To see the Review answers, open this PDF file and look for section 12.9.

Resources

 

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Vocabulary

Favorable Outcome

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

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

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

Mutually Exclusive Events

Mutually exclusive events have no common outcomes.

Sample Space

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

Total Outcomes

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

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.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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