# ﻿Keeping Track of All Possible Outcomes

Warmup

Four friends; Amy, ben, Chloe, and Dan send out Christmas cards to each other. See the applet below.

How many cards were sent out among the four friends?

Activity #1

Introduction to Tree Diagrams.

One of the easiest ways to solve a probability problem is to construct the probability tree and then read the answer. Now, a probability tree diagram is a diagram that is used to give a visual representation of the possibilities as well as the outcomes of an event. It also indicates some conditional probabilities for combinations of two or more events.

• Read the story below and follow to diagram to see how a tree diagram in probability is created.

Noah is planning his birthday party. Here is a tree showing all of the possible themes, locations, and days of the week that Noah is considering.

(1.) How many themes is Noah considering?

(2.) How many locations is Noah considering?

(3.) How many days of the week is Noah considering?

(4.) One possibility that Noah is considering is a party with a space theme at the skating rink on Sunday. Write two other possible parties Noah is considering.

(5.) How many different possible outcomes are in the sample space?

Activity #2

Exploring two-step Tree Diagrams.

Susan’s breakfast consists of two types of main dishes and 2 types of drinks. See the possible choices she can make on the right side of the applet below.​

• Increase the number of main dishes to three and note how many choices she can now make.​
• Increase the main dishes to four and again note how many choices she can make.
• Click the “Reset” button at the top right corner and increase number of drinks to 3, to four noting the number of choices she can make.
• Repeat the steps until you have four of each type.

Activity #3

`Listing Sample Space for a Compound Event.`

If an event has more than one possible outcome, it is termed as a compound event. For example flipping a coin and rolling a die. Compound events are a little more complex than simple events. These events involve the probability of more than one event occurring together.

Consider the experiment: Flip a coin, and then roll a number cube.

Elena, Kiran, and Priya each use a different method for finding the sample space of this experiment.

• Elena carefully writes a list of all the options: Heads 1, Heads 2, Heads 3, Heads 4, Heads 5, Heads 6, Tails 1, Tails 2, Tails 3, Tails 4, Tails 5, Tails 6.
• Kiran makes a table:

Priya draws a tree with branches in which each pathway represents a different outcome:

(1.) Compare the three methods. What is the same about each method? What is different? Be prepared to explain why each method produces all the different outcomes without repeating any.

(2.) Which method from the previous question do you prefer for this situation?

Challenge #1

There is a bag of 50 marbles.

• Andre takes out a marble, records its color, and puts it back in. In 4 trials, he gets a green marble 1 time.
• Jada takes out a marble, records its color, and puts it back in. In 12 trials, she gets a green marble 5 times.
• Noah takes out a marble, records its color, and puts it back in. In 9 trials, he gets a green marble 3 times.

Estimate the probability of getting a green marble from this bag. Explain your reasoning.

Challenge #2

A simulation is done to represent kicking 5 field goals in a single game with a 72% probability of making each one. A 1 represents making the kick and a 0 represents missing the kick.

Challenge #3

For the event event below, write the sample space and tell how many outcomes there are.

Lin selects one type of lettuce and one dressing to make a salad.

• Lettuce types: iceberg, romaine
• Dressings: ranch, Italian, French

Quiz Time