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# Tree Diagrams

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Practice Tree Diagrams
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## Tree Diagram

### Topic

Representing Sports Outcomes with Discrete Random Variables and Tree Diagrams

### Vocabulary

• Random variables
• Discrete random variables
• Tree Diagram
• Probability
• Experimental Probability

### Student Exploration

#### Which sports outcomes can be represented with discrete random variables? How can you use a tree diagram to represent the possible outcomes?

Sports!

In most sports there are many different elements of the games that can be represented with discrete random variables. For each of the following sports situations create a tree diagram. A tree diagram is a way to show the outcomes of simple probability events, where each outcome is represented as a branch on the tree. If you need help creating the tree diagrams, revisit the lesson on tree diagrams.

Below are examples of the tree diagrams for #1 and 2.

1. Free throws in basketball. Make a tree diagram representing all the possible outcomes of shooting three free throws.

2. Penalty shoot-out kicks in soccer. Make a tree diagram representing all of the possible outcomes of shooting five shoot-out penalty kicks.

3. Sets won in a tennis match. Make a tree diagram representing all of the possible outcomes for two players in a tennis match that is in a best of five sets system.
4. Shots in a Hockey shoot-out. Make a tree diagram representing all of the possible outcomes of shooting three penalty shoots.
5. Why is it that we can use tree diagrams to represent situations with discrete random variables? And we cannot create tree diagrams for situations that cannot be represented with discrete random variables? Explain each!
(As a reminder, discrete random variables represent the number of distinct values that can be counted of an event, for more information revisit the lesson on discrete random variables).
Situations with discrete random variables can be represented with tree diagrams because there are a specific number of outcomes in the event and each of those outcomes can be represented with a branch in the tree diagram. Where as with situations that cannot be represented with discrete random variables, there are not a specific number of outcomes, and therefore it cannot bedetermined how many branches to make in the tree diagram.

Resources for Sports Rules:

6. Choose one of the situations from above and conduct your own experiment to determine your experimental probability of completely that situation successfully. Use proper probability notation to express the probability of you completing that task successfully.

Answers will vary but here is an example. $P(\text{scoring penalty kick}) = \frac{3}{5}$.

Tennis player.

### Extension Investigation

Many situations cannot be represented by a discrete random variable. Explain why each of these situations cannot.

1. The hitting options for one baseball or cricket player inone round at bat. (Need a hint? search for longest “at bat”). Technically a baseball player could be at bat for an unknown number of times. Even though a batter is given three strikes (where they attempt to hit the ball) and four balls (where the pitcher makes errors) because when they hit a foul ball it counts as a strike but they can’t get out on a foul ball. So the number of hits is unknown. Although there is not official record for the longest session at bat, there are many sports fans that have their commentary online see: http://www.sptimes.com/2006/04/13/Rays/The_longest_AB_We_hav.shtml or http://answers.yahoo.com/question/index?qid=20070420141850AAJlrox or http://en.allexperts.com/q/Baseball-Trivia-General-2552/longest-time-bat.htm.
2. The in and out serving options in a volleyball game.
3. The number of completed passes in an American football game.
4. The number of strokes made in a hole of golf.
5. Choose one of the situations from #7 – 10 and explain why it also cannot be represented with a tree diagram. You could not represent it with a tree diagram because the number of possible outcomes is unknown. There could be any number of strokes in on hole of golf.
6. Sometimes there might be too many outcomes to feasibly make a tree diagram. Give an example of this type of situation. What other methods might someone use to organize his or her work when a tree diagram is not useful?

Resources for Sports Rules:

### Connections to other CK-12 Subject Areas

• Tree Diagrams
• Discrete Random Distributions
• Probability and Permutations
• Theoretical and Experimental Probability