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Fundamental Counting Principle with and without Repetition

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Calculating Outcomes
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Have you ever been to a bike shop? Take a look at this dilemma.

Telly and Carey are working in a bicycle shop over the spring vacation. They are excited because Ms. Kelley the owner is going to let each of them choose a new bike when the week is done. This way, they get to work and earn a new bike at the same time.

There are so many different types of bikes, that Telly is having a difficult time choosing. To complicate matters even more, Ms. Kelley said that she would let Telly design her own bike so she can choose seat color and type of handlebars as well as color of the bike. Telly knows that she wants a mountain bike, so at least that part has been chosen. Here are all of the options that Telly has.

Mountain bike

Colors = Red, Green, Blue or Purple

Seat = normal or extra cushion

Handlebars = straight or curved

Telly took out a piece of paper and tried drawing out all of her options. She became frustrated almost immediately.

This is where you come in. You can help Telly to work through this problem by learning about the Counting Principle. Pay attention because you will see this problem again.

Guidance

Probability is a mathematical way of calculating how likely an event is likely to occur.

An event is a result of an experiment or activity that might include such things as:

  • flipping a coin
  • spinning a spinner
  • rolling a number cube
  • choosing an item from a jar or bag

An important concept when calculating probability is to think about outcomes.

An outcome is a possible result of some event occurring. For instance, when you flip a coin, “heads” is one outcome; tails is a second outcome.

Total outcomes are computed simply by counting all possible outcomes.

That is a great question.

We can use arithmetic to calculate outcomes.

Take a look at this situation.

Ichiro’s Car Wash offers three different wash services – basic outer body wash, interior cleaning, and finally, custom hand-detailing. Customers order a wash choice plus wax or no wax. You can use a tree diagram to find that there are 6 different choices, or outcomes, for a car wash.

We can look at the situation in terms of outcomes. For the first choice there are 3 different outcomes. For the second choice there are 2 different outcomes.

3 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 6 \ \text{outcomes}

What if we changed the options? Would this method still work? Let’s add on to the car wash problem and see.

Consider Ichiro’s oil change choices. Customers can get standard or synthetic oil, filter or no filter, and pay with a coupon or for the regular price. For each choice there are 2 different outcomes.

2 \ \text{outcomes} \cdot 2 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 8 \ \text{outcomes}

This method of calculating the number of total outcomes can be stated as a general rule called the Counting Principle .

Counting Principle : The number of choices or outcomes for two independent events, A and B taken together, is the product of the total number of outcomes for each event.

 \text{Total outcomes for} \ A \ \text{and} \ B = \text{(number of outcomes for} \ A) \cdot \text{(number of outcomes for} \ B)

Once again, the Counting Principle requires that we take the number of choices or outcomes for two independent events and multiply them together. The product of these outcomes will give us the total number of outcomes for each event.

For instance, 2 tosses of a coin there are 2 outcomes for each toss. Using the Counting Principle, you can find the total number of outcomes as:

2 \ \text{outcomes} \cdot 2 \ \text{outcomes} = 4 \ \text{total outcomes}

You can use the Counting Principle to find probabilities of events. For example, suppose you wanted to know the probability of rolling two number cubes and coming up with a sum of 5. The probability of any event is equal to the ratio of favorable outcomes to the total number of equally likely possible outcomes.

& P \text{(event)} = \frac{\text{favorable outcomes}}{\text{total outcomes}}

Favorable outcomes are the outcomes you are looking for.

In this case the favorable outcomes are outcomes that have a sum of 5. To find the number of total outcomes for the two tosses, you can use the Counting Principle. Since each toss of a number cube has 6 different outcomes.

 \text{total outcomes} &= 6 \ \text{outcomes} \cdot 6 \ \text{outcomes}\\&= 36\ \text{total outcomes}

To figure out this particular problem we must add another step. Now list those 36 outcomes and mark the outcomes that result in a sum of 5.

1-1 \quad 2-1 \quad 3-1 \quad {\color{red}\mathbf{4-1}} \quad 5-1 \quad 6-1 \\1-2 \quad 2-2 \quad {\color{red}\mathbf{3-2}} \quad 4-2 \quad 5-2 \quad 6-2 \\1-3 \quad {\color{red}\mathbf{2-3}} \quad 3-3 \quad 4-3 \quad 5-3 \quad 6-3 \\{\color{red}\mathbf{1-4}} \quad 2-4 \quad 3-4 \quad 4-4 \quad 5-4 \quad 6-4 \\1-5 \quad 2-5 \quad 3-5 \quad 4-5 \quad 5-5 \quad 6-5 \\1-6 \quad 2-6 \quad 3-6 \quad 4-6 \quad 5-6 \quad 6-6

Since there are 4 outcomes that have a sum of 5:

P(5) = \frac{4}{36}=\frac{1}{9}

The probability of rolling two number cubes with a sum of 5 is \frac{1}{9} .

Calculate each outcome.

Example A

Jake has two shirts, four pairs of pants and three sweaters. How many different outfits can he make?

Solution: 24 outcomes

Example B

What if Jake has two shirts, five pairs of pants and four sweaters. How many outfits can he make?

Solution: 40 outcomes

Example C

What if Jake only chooses one sweater. How many outfits can he create with four pairs of pants and two shirts.

Solution: 8 outcomes

Now let's go back to the dilemma from the beginning of the Concept.

First, list out the options once again.

Mountain bike

Colors = Red, Green, Blue or Purple

Seat = normal or extra cushion

Handlebars = straight or curved

Next, we can use the Counting Principle to calculate the total number of options that Telly has for her bike.

There are four colors = 4

There are two seat options = 2

There are two handlebar options = 2

4 \times 2 \times 2 = 16 possible bike options

This is the answer.

Vocabulary

Probability
the mathematical way of calculating the likelihood of an event occurring, the ratio of favorable outcomes to total outcomes.
Event
a result of an experiment or activity.
Outcome
a possible result of an event occurring.
Total Outcomes
all possible outcomes
Counting Principle
 \text{Outcomes} \times \text{outcomes} = \text{total outcomes}
Favorable Outcomes
The outcomes that you are looking for.

Guided Practice

Here is one for you to try on your own.

What is the probability of spinning the spinner twice and having it land on the same color both times?

Solution

Step 1 : Rather than draw a tree diagram, use the Counting Principle to find the number of total outcomes. Since each spin has 4 outcomes:

 \text{total outcomes}&= 4 \ \text{outcomes} \cdot 4 \ \text{outcomes}\\ &= 16 \ \text{total outcomes}

Step 2 : Now list all 16 outcomes and find the number of ways both spinners can land on the same color both times:

&{\color{red} \mathbf{red - red}} && \text{blue}-\text{red} && \text{yellow}-\text{red} && \text{green}-\text{red} \\&\text{red}-\text{blue} && {\color{red} \mathbf{blue - blue}} && \text{yellow}-\text{blue} && \text{green}-\text{blue}\\&\text{red}-\text{yellow} && \text{blue}-\text{yellow} && {\color{red} \mathbf{yellow - yellow}} && \text{green}-\text{yellow}\\&\text{red}-\text{green} && \text{blue}-\text{green} && \text{yellow}-\text{green} && {\color{red} \mathbf{green - green}}

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

 P \text{(same)} = \frac{favorable}{total \ outcomes}=\frac{4}{16}= \frac{1}{4}

The probability of the arrow landing on the same color two times in a row is \frac{1}{4} .

Video Review

The Counting Principle

Practice

Directions: Use the counting principle to determine the number of outcomes.

  1. Nigel dropped 3 open-faced peanut butter sandwiches that were equally likely to land face-up or face-down. How many different outcomes are there?
  2. Movie Star Toothpaste comes in 3 different flavors: sparkle, blast, and stripe, 3 different sizes, and 2 different tube styles. How many toothpaste choices are there?
  3. Dave made 3 predictions for this Sunday’s football games. How many different outcomes of being right or wrong are there?
  4. Bridget made 4 predictions for this Sunday’s football games. How many different outcomes of being right or wrong are there?
  5. How many more outcomes will there be if Bridget adds an extra prediction for problem 4 above?
  6. Sandy’s Sandals come in 4 different models: sport, super-sport, casual, and chic, 5 different colors, and 9 different sizes. How many choices are there?
  7. Mike’s Bikes features three different bike styles: mountain, racer, and stunt bike. You can choose from 6 different gear systems, 4 frame alloys, and 5 colors. How many bike choices are there?
  8. In problem 8 above, Mike gets a new bike color but now features only 5 different gear systems. Does he have more or fewer choices now? How many more or fewer?
  9. Tilly spins a spinner that has red, blue, and yellow sections 3 times. How many different outcomes are possible?
  10. Tilly spins a spinner that has red, blue, and yellow sections and tosses a number cube. How many different outcomes are possible?
  11. Tilly tosses a number cube 2 times. How many different outcomes are possible?
  12. Tilly tosses a number cube 3 times. How many different outcomes are possible?

Directions: Use the counting principle to determine probabilities for a spinner with four colors: red, yellow, blue and green.

  1. Kalyani spins the spinner 2 times. What is the probability that the arrow will land on green both times?
  2. Kalyani spins the spinner 2 times. What is the probability that the arrow will land on the same color both times?
  3. Kalyani spins the spinner 2 times. What is the probability that the arrow will land on a different color each time?

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