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12.11: Counting Rule to Calculate Probabilities

Difficulty Level: At Grade Created by: CK-12
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Practice Counting Rule to Calculate Probabilities

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Remember the talent show? Take a look at this dilemma.

Jacob is going to do a magic trick at the talent show. He is going to put four scarves into a hat. The scarves are red, yellow, blue and green. If he does this trick twice, what are the chances that he will pull out a green scarf both times?

In this Concept you will learn how to calculate this probability. Pay attention and you will see this again at the end of the Concept.

Guidance

You can use the Counting Principle to find probabilities of events. For example, suppose you wanted to know the probability of the arrow landing on the same color on both spinners. Keep in mind that for any probability you can use this ratio.

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

Here, you can use the Counting Principle to find the number of total outcomes for the two spins with the two spinners. There are four outcomes on one spinner and three outcomes on the other spinner.

\begin{align*}\text{Total outcomes} &= 4 \ \text{outcomes} \cdot 3 \ \text{outcomes}\\ &= 12 \ \text{outcomes}\end{align*}

Now list those 12 outcomes and mark the outcomes that are the same color for both spins.

\begin{align*}& \text{red-red} && \text{blue-red} && \text{yellow-red} && \text{green-red}\\ & \text{red-blue} && \text{blue-blue} && \text{yellow-blue} && \text{green-blue}\\ & \text{red-green} && \text{blue-green} && \text{yellow-green} && \text{green-green}\end{align*}

Since there are 3 outcomes that have the same color for both spins:

\begin{align*}P (\text{same}) = \frac{3}{12}=\frac{1}{4}\end{align*}

The probability of both spinners landing on the same color is \begin{align*}\frac{1}{4}\end{align*}.

You can also apply the Counting Principle to a variety of different probability problems.

Example

Anna flips a coin 3 times in a row. What is the probability that she will get heads all 3 times?

Step 1: Rather than draw a tree diagram, to find the number of total outcomes you can simply multiply the number of outcomes for each flip

\begin{align*}\text{Total outcomes} &= 2 \ \text{outcomes} \cdot 2 \ \text{outcomes} \cdot 2 \ \text{outcomes}\\ &= 8 \ \text{outcomes}\end{align*}

Step 2: Now list all 8 outcomes and find the number of ways Anna can get heads all 3 times. Clearly, there is only one arrangement in which all 3 flips result in heads.

\begin{align*}& \text{heads-heads-heads} && \text{tails-heads-heads}\\ & \text{heads-heads-tails} && \text{tails-heads-tails}\\ & \text{heads-tails-heads} && \text{tails-tails-heads}\\ & \text{heads-tails-tails} && \text{tails-tails-tails}\end{align*}

Step 3: Find the ratio of favorable outcomes to total outcomes:

\begin{align*}P (\text{red-red-red}) = \frac{1}{8}\end{align*}

Use the Counting Principle to find each probability.

Example A

Abra flips a coin 2 times. What is the probability that both flips will match?

Solution: \begin{align*}\frac{2}{4} = \frac{1}{2}\end{align*}

Example B

Abra flips a coin 2 times. What is the probability that both flips will NOT match?

Solution: \begin{align*}\frac{2}{4} = \frac{1}{2}\end{align*}

Example C

Abra flips a coin 3 times. What is the probability that all 3 flips will match?

Solution: \begin{align*}\frac{2}{8} = \frac{1}{4}\end{align*}

Here is the original problem once again.

Jacob is going to do a magic trick at the talent show. He is going to put four scarves into a hat. The scarves are red, yellow, blue and green. If he does this trick twice, what are the chances that he will pull out a green scarf both times?

There are four possible outcomes in this trick.

Jacob will perform the trick twice.

\begin{align*}2 \times 4 = 8\end{align*}

There are 8 outcomes possible.

Two of these outcomes could have the scarf chosen be green.

\begin{align*}\frac{2}{8} = \frac{1}{4}\end{align*}

Vocabulary

Here are the vocabulary words used in this Concept.

Probability
the ratio of favorable outcomes to total possible outcomes.
The Counting Principle
the product of the outcomes of a series of events gives the total number of outcomes.

Guided Practice

Here is one for you to try on your own.

If Sam flips a coin five times. What is the probability of having it come up heads all at the same time?

First, let's figure out the total number of outcomes.

There can be two possible outcomes each time Sam flips the coin.

\begin{align*}2 \times 5 = 10\end{align*}

There are 10 possible outcomes.

This can only happen one time.

Our answer is \begin{align*}\frac{1}{10}\end{align*}.

Video Review

Here is a video for review.

Practice

Directions: Find probabilities using the Counting Principle.

1. Abra flips a coin 3 times. What is the probability that all 3 flips will NOT match?

2. Abra flips a coin 3 times. What is the probability that heads will come up exactly 2 times?

3. Billy spins the spinner twice. What is the probability that blue will come up both times?

4. Billy spins the spinner twice. What is the probability that blue will come up at least one time?

5. Billy spins the spinner twice. What is the probability that blue will come up exactly one time?

6. Billy spins the spinner twice. What is the probability that both spins will match?

7. Cindy tosses a number cube two times. What is the probability that both tosses will match?

8. Cindy tosses a number cube two times. What is the probability that both tosses will NOT match?

9. Cindy tosses a number cube two times. What is the probability that the sum of the two tosses will be 7?

10. Cindy tosses a number cube two times. What is the probability that the sum of the two tosses will be greater than 7?   11. Tyler rolls a number cube three times. What is the probability that he will roll a two?

12. Tyler rolls a number cube three times. What is the probability that he will roll an odd number?

13. What is the probability that he will roll an even number?

14. If Tyler rolls a number cube once, what is the probability of rolling a two or a five?

15. If Tyler rolls a number cube six times, what is the probability of rolling a two or a five?

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Color Highlighted Text Notes

Vocabulary Language: English

TermDefinition
Probability Probability is the chance that something will happen. It can be written as a fraction, decimal or percent.
The Counting Principle In probability, the counting principle states that the number of outcomes for two independent events taken together is the product of the total number of outcomes for each individual event.

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