6th Grade - Show Data Using Tables And Tree Diagrams

 
     
 
     
 
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6th
Statistics
Show Data Using Tables and Tree Diagrams
Represent all possible outcomes for compound events in an organized way (e.g., tables, grids, tree diagrams) and express the theoretical probability of each outcome.
Understand that a compound event is a combination of two or more single events. The event is the result or outcome of an experiment. Find all possibilities (outcomes) for an experiment that includes more than one part (event) in an organized way. Create a tree diagram, an organized list/table, grid or use the Fundamental Counting Principal to count and show all possibilities. Use a fraction, decimal or percent to represent the theoretical probability of an event. Know that the probability of an event is the likelihood that the event will occur, where 0 shows that and event is impossible and 1 shows that it is certain. All other probabilities can be written as a fraction, decimal and percent. Understand that theoretical probability is shown as a ratio with the numerator showing favorable outcomes, the possibility of a specific event; and the denominator showing the sample space, set of all possible outcomes together as one.
 

Sample Problems

(1)

A man has 3 shirts: red, black, and blue and 2 pairs of pants: jeans and slacks. Make a tree diagram to show all possible ways of combining the pants with the shirts. What is the probability that the man will wear the red shirt with the jeans?

(Sample Tree Diagram:

red


jeans black

blue



red


slacks black

blue


The probability that the man will wear the jeans and red shirt is 1/6.)


(2)

Make an organized list of all possible outcomes for flipping a penny, nickel and a dime. What is the probability that all coins will land on heads?

(Sample list:

Penny Nickel Dime

T T T

T T H

T H T

T H H

H T T

H T H

H H T

H H H

The probability that all coins will land on heads is 1/8.)

(3)

Will can buy a lunch combination with a small, medium or large drink; sandwich or burger for the main course and side of either fruit or salad. Find the total number of all possible combinations of his lunch if he includes a drink, a main course and a side. Explain your answer

(12 possible combinations. The Fundamental Counting Principal can be used because you need to find the total number of all possible combinations. There are 3 drink choices, 2 main course choices and 2 side choices. 3 x 2 x 2 = 12.)

(4)

Use a grid to show all possible outcomes if your spin a 4 section spinner numbered 1, 2, 3, 4 two times. Use the grid to find the probability that you will spin the combination of a 3 on the first spin and then a 4 on the second spin.

(Sample Grid:


1

2


3


4

1

1,1

1,2

1,3

1,4

2

2,1

2,2

2,3

2,4

3

3,1

3,2

3,3

3,4

4

4,1

4,2

4,3

4,4

The probability of spinning a 3,4 is 1/16.)


(5)

Use a grid from exercise 4 to show all possible outcomes if your spin a 4 section spinner numbered 1, 2, 3, 4 two times. Use the grid to find the probability that you will spin the combination of a 1 and a 4 together in any order.

(The probability of spinning a 1 and 4 together is 2/16 or 1/8.)


Learning Tips

(1)

Be sure that your child understands that a compound event is a combination of two or more single events. You may want to help your child to understand the meaning of compound by relating it to compound words. Remind your child that compound words are two words that are put together as one (i.e. baseball, goldfish, fingernail, paperclip, etc…). Compound events can either be the same event repeated two or more times (i.e. roll a die two times) or two different events going on at the same time (i.e. roll a die and spin a spinner). The event is any outcome, or result, of a single experiment.

(2)

Children need to understand the theoretical probability of an event occurring. Probabilities range from 0 to 1, where 0 is impossible for an event to occur and 1 is 100% certain that an event will occur. Probabilities can be used to tell whether an event is; certain, very likely, equally likely, less likely and impossible to occur. The chart below shows how each probability can be labeled based on its likeliness.

Certain

1/1 or 100%

Very likely/More likely

Any fraction more than ½, but less than 1. 51% to 99%

Equally likely

½ or 50%

Less likely

Any fraction less than ½, but more than O. 49% - 1%

Impossible

0/1 or 0%


Probabilities can be written as a fraction, a decimal or a percent. You may want to discuss how we hear probabilities in the media with weather forecasts, etc…


It is important that your child understand that when he/she finds theoretical probability, that he/she must always compare the number of favorable outcomes (probability being sought out) to the total number of possible outcomes (total possible combinations or choices). It may help to discuss the meaning of the word theoretical or even break it down into the base word, theory. Remind you child that a theory is an educated guess about what could happen. This is why when we look at theoretical probability, we don’t actually try the experiment, we just think about all possible outcomes that could happen if we did.


The probability of an event is shown a P(n), where n is replaced with the event that is being solved for. So, if we were to be solving for the chances of spinning red on a spinner, we would write P(red) or P(r). Theoretical probability can be shown as a fraction. The numerator (top number) of the fraction will always be the number of favorable outcomes. The denominator (bottom number) of the fraction will always be the number of possible outcomes. The theoretical probability for any event would be shown as; P(event) = number of favorable outcomes

number of all possible outcomes

The words would be replaced with numbers to solve for a specific probability. Here’s an example of how to use the formula above to find the theoretical probability of rolling a even number on a number cube 1-6.

  1. Write the probability you are seeking in the parenthesis after the P.

P (even)

  1. Count the possible favorable outcomes. The favorable outcomes for this problem are all even numbers on the cube. 2, 4, and 6 are even. Hence, there are 3 favorable outcomes of even numbers.

P(even) = 3__

  1. Count all possible outcomes. There are 6 numbers on the cube, which means a total of 6 possible outcomes.

P(even) = 3__

6

  1. If possible, write the fraction in simplest form by dividing both the numerator and denominator by the greatest common divisor (largest number that will divide into both evenly).

3/6 are both divisible by 3. So, the theoretical probability in simplest form is 1/3.

(3)

It is sometimes necessary to find all possible outcomes for a compound event. All possible outcomes are the total possibilities for all combinations of events together. For example, if you’re asked to find all possible outcomes for rolling a die 4 times, you’re actually being asked how many combinations are there for all the numbers that could possible appear each time. Or, if you’re asked to find how many possibilities of outfits Seana has if she has 10 skirts, 5 tops, and 4 pairs of shoes, you’re being asked to find the total outfits she has if she combines all skirts, tops and shoes in every possible way. The easiest way to find all possible outcomes is to use the Fundamental Counting Principal. The Fundamental Counting Principal allows you to multiply the total number of choices for each event (roll, spin, size, design, style, etc…) to find the total possible outcomes, or choices. In order to use this principal, you’ll need to be sure to always multiply and include the total of each event. Here’s how you would use it for each of the examples above.

Sample 1: What are the total possible outcomes if you roll a die numbered 1-6 four times?

  1. Find the total number of events. (4)

  2. Find the total outcome for each event. (6, because there are 6 different numbers on the cube.

  3. Multiply the total outcome for each event by the total outcome of the next event.

Roll 1 Roll 2 Roll 3 Roll 4

6 x 6 x 6 x 6

4. Calculate the total possible outcomes by multiplying all numbers.

6 x 6 = 36 x 6 = 216 x 6 = 1296 number combinations


Sample 2: Seana has if she has 10 skirts, 5 tops, and 4 pairs of shoes, you’re being asked to find the total outfits she has if she combines all skirts, tops and shoes in every possible way.

  1. List each of the events (items to choose) and their total choice amounts.

10 skirts 5 tops 4 shoes

2. Multiply the total of all events together.

10 x 5 = 50 x 4 = 200 combinations of outfits


For the visual/kinesthetic learner, you may want to show how the Fundamental Counting Principal works using clothing, food choices or other items from your home. The kinesthetic learner can make different meal combinations by using the actual food or clothing item. The visual learner can draw pictures of all the possible combinations.

(4)

Creating a table or list is another way to show all possible outcomes for a compound event. Tables and lists are especially helpful when you’re asked to find a specific probability. When creating a list or inputting possible outcomes into a table it is most important to list all possible outcomes in an organized way so that no events are left off or repeated. This is why you should work on all possible outcomes one event at a time. Here’s an example. A store sells red and blue hats in sizes small, medium, and large. If Tony randomly chooses a hat from the rack without looking, what are the chances it will be a red hat, size medium? This problem is asking for the specific probability of choosing a red, medium hat. You will not be able to find this out until you know the sample space, or the total possible outcomes. All possibilities are shown in the list below.

Red Blue

Sm. Sm.

Med. Med.

Lg. Lg.

The total possibilities could also be shown in a table like the one below.

Red

small

medium

large

Blue

small

medium

large

Either organization visual will give you the answers you need. It is important that each learner decide for him/herself which method is easiest to comprehend. Both visuals show that there are a total of 6 possible outcomes. We are asked to find the probability of choosing a red hat, size medium. We can see by looking at the list or table that there is one choice of a hat that is medium and red. Red, medium is 1 choice out of 6 possible choices. So, Tony has the theoretical probability of 1/6 chances of choosing a red, medium hat.

Lists can be written vertically like the list above or horizontally. The table is horizontal and could be written vertically. It is up to each individual to create different kinds of lists and tables to determine what will work best for them.

Place different items from your home in a pile and have your child create neat lists and/or tables to show all possible outcomes for closing their eyes and choosing that item.

(5)

A grid can be helpful in showing all possible outcomes for repeating an event two times. The numbers, letters, or word going across the top represent all possible outcomes, as do the numbers on the far left side. The remaining pairs of numbers, letters or words show possible outcomes after both events have occurred. A problem where grid could be helpful would be: Danny spun a spinner two times. Each section of the spinner had the letters s-p-i-n. What is the probability that Danny will get the letter s on both spins?

To create your grid, you need to determine how many rows and columns you will need. There are 4 letters on the spinner, but you will need 5 rows and 5 columns in your grid so that you have room to show the letters across the top of the grid and going along the side from top to bottom. Begin your grid by setting this up, as seen below.


S

P

I

N

S





P





I





N





Next, fill in the middle of the chart by first listing the letter shown on the row on the left and the column above. An example is shown below.


S

P

I

N

S





P



P, I


I





N





The P was listed first, because the P is the row, I is second because this is the column. The same pattern should be used to fill in the remaining grid items. (You could choose to list the letters vertically by column, but you would just need to keep this consistent in all columns. Choose a pattern and stick with it.)


S

P

I

N

S

S, S

S, P

S, I

S, N

P

P, S

P, P

P, I

P, N

I

I, S

I, P

I, I

I, N

N

N, S

N, P

N, I

N, N

The visual learner may need to use color-coding to keep the actual possibilities separate from the events. This can be done by highlighting all the pairs or using a different colored pencil for the original event.


S

P

I

N

S

S, S

S, P

S, I

S, N

P

P, S

P, P

P, I

P, N

I

I, S

I, P

I, I

I, N

N

N, S

N, P

N, I

N, N

Now the grid can be used to find the probability requested. If we need to find the probability that Danny can get an “s” on both spins, we need to find that on the chart as S, S. There is one S,S out of a total of 16 pairs. Thus, the probability of S,S is 1/16.

You can have your child practice creating a grid for a die that is tossed 2 times.

(6)

Tree diagrams are another way to organize all possible outcomes for compound events or name specific probabilities. Below is an example of how a tree diagram can be used to show all possibilities for tossing a coin (heads or tails) and a number cube (1-6).

It is easiest to use the least number of possibilities as your start point. Since the coin only has two possibilities: heads or tails, we will start here.


Heads




Tails


Next, we will think about the outcome of all possible outcomes for tossing the number cube. We will add all these possibilities to both heads and tails.

1

2

Heads 3

4

5

6


1

2

Tails 3

4

5

6


We can use the tree diagram to find all possibilities by counting all possibilities shown when the diagram is completed. In this case, there are 12 possible combinations of the coin and the number cube.


A tree diagram can also be used to find a specific theoretical probability. For example, if we needed to find the probability of getting tails and a 3, we would look at the tree diagram for tails and find how many 3s there are. There is only one 3 with tails, which means one possible outcome. To find the sample space we will need to use the number of all possible combinations, 12. So, we have a 1/12 chance of getting tails and 3.


Visual learners may want to draw heads and tails instead of writing the words. They can also show the dots or numbers on the number cube.


Kinesthetic learners will best be able to create the tree diagrams if they have the actual experiment times to look at, hold, and investigate. For example, you can have them use a spinner from a board game and a die. They will then be able to create a tree diagram for each part of the spinner with each dot on the die.

Extra Help Problems

(1)

Write the theoretical probability of rolling an odd number on a cube labeled 1-6. Describe the likeliness of this event.

(1/2, equally likely)

(2)

What is the theoretical probability of pulling and “E” out of a box that has tiles with the letters: T-H-E-O-R-E-T-I-C-A-L. Describe the likeliness of this event.

(2/11, less likely)

(3)

What is the theoretical probability of pulling the name of a day of the week with the letter “a” in it, if a hat slips of papers with each day of the week inside.

(7/7, 1, It is certain, because all days of the week have at least one “a”.)

(4)

Write the theoretical probability of rolling a 7 number on a cube labeled 1-6. Describe the likeliness of this event.

(0/7, impossible)

(5)

A spinner has 4 sections colored: red, yellow, red, and red. What is the theoretical probability of landing on red on your first spin? Describe the likeliness of this event.

(3/4, more likely)

(6)

A spinner has 4 sections colored: red, yellow, red, and red. What is the theoretical probability of landing on yellow on your first spin? Describe the likeliness of this event.

(1/4, less likely)

(7)

Tanya has a 7 colored pencils, 3 colors of highlighters and 4 pens with different inks. What is her total of all possible combinations of these writing utencils?

(84 possible outcomes)

(8)

Jake has 4 ties, 7 shirts and 5 pairs of shoes. How many different outfits can he make if he mixes and matches all ties, shirts and shoes?

(140 outfits)

(9)

Amanda has 12 dresses, 5 necklaces, and 11 pairs of sandals. How many total choices does Amanda have for what to wear to the school dance?

(660 different combinations)

(10)

Roy is at a deli of lunch. The lunch special offers 5 main course meals, 4 drink options and 9 kinds of chips. How many different lunch combinations does Roy have to choose from?

(180 possible combinations)

(11)

A store sells plain t-shirts in red, yellow, blue, pink and white. They have sizes: small, medium, large and extra large. Make an organized list to show all possibilities plain shirt choices.

(Sample List:

red: small, med., lg., x-lg.

yellow: small, med., lg., x-lg.

blue: small, med., lg., x-lg.

pink: small, med., lg., x-lg.

white: small, med., lg., x-lg.)

(12)

Use your list from exercise 11 to find the probability of randomly choosing a pink, small shirt without looking.

(1/20)

(13)

A restaurant has a kid’s meal with the choice of a main course of a hamburger, hotdog or chicken strips. The meal comes with a side of their fruit or fries. The drink choices include: milk or juice.

Draw a tree diagram to show the total possible different meal combinations kids have to choose from.

(Sample tree diagram:



milk

fruit

hamburger juice

milk

fries

juice







milk

fruit

hotdog juice

milk

fries

juice





milk

fruit

chicken juice

strips

milk

fries

juice




(14)

Use your tree diagram from problem #14 to determine all possible outcomes for combinations of meals. Check your answer using the Fundamental Counting Principal.

(12 possible combinations, 3 x 2x 2 = 12)

(15)

Use the tree diagram from problem #14 to determine the probability that Johnny will receive a hamburger, fries and juice if he randomly points to the menu without looking.

(1/12)

(16)

Stephanie a spinner with 4 sections colored: red, blue, green and orange. Create a grid to show all possibilities if Stephanie spins two times.

(Sample grid:


red

blue

green

orange

red

r,r

b, r

g, r

o, r

blue

r,b

b, b

g, b

o, b

green

r,g

b,g

g, g

o, g

orange

r,o

b, o

g, o

o, o

(17)

Use the grid from problem 16 to find all possible outcomes.

(16)

(18)

Use the grid from problem 16 to find the theoretical probability of spinning an o and a g in any order.

(2/16, 1/8)

(19)

Tori is playing a came with a number cube labeled 1-6 and a quarter. Use a visual of your choice to show all possible outcomes for these events. Then use your visual to determine the probability of Tori landing on heads and 6.

(1/12)

(20)

Mike is spinning a spinner with 5 letters in each section: M-A-T-H-A. Draw a visual to show all possible outcomes. Then use your visual to find the probability of Mike getting the letter A both times if he spins twice.

(4/25)

(21)

Ethan is pulling colored cards off of a pile with: 1 queen, 3 kings, and 2 jacks. He also has a spinner numbered 1-8. Create a visual of your choice to show all possible outcomes. Use your visual to tell the theoretical probability of Ethan drawing king with an even number.

(12/48, ¼)

(22)

Ethan is pulling colored cards off of a pile with: 1 queen, 3 kings, and 2 jacks. He also has a spinner numbered 1-8. Create a visual of your choice to show all possible outcomes. Use your visual to tell the theoretical probability of Ethan drawing Jack and a 1.

(1/24)

(23)

Bekka is playing a game with a die. If she rolls it 3 times, what will her probability of rolling only 2s and/or 3s be? Draw a visual of your choice to show all possibilities.

(6/18, 1/3)

(24)

Bekka is playing a game with a die. If she rolls it d times, what will her probability of rolling a total of 11 be? Draw a visual of your choice to show all possibilities.

(1/6)

(25)

Juan is flipping a penny, nickel and quarter. What is the probability that all three coins will land on tails?

(1/8)


 

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