Gamers
Rationale
This task provides a context for using a two-way frequency table and a Venn diagram to explore the relationships among probabilities.
Instructional Task
JBobby and Esmeralda, two avid video game players, found a list of the top 100 video games of all time. They discovered that the list had many titles they did not know. The students created a two-way frequency table (also known as a contingency table) displaying the number of games on the list that they had played.
| Esmeralda played | Esmeralda did not play | Total | |
|---|---|---|---|
| Bobby played | 6 | 14 | 20 |
| Bobby did not play | 4 | 76 | 80 |
| 10 | 90 | 100 |
Part I. Determine the probability that if a game was selected at random from the list—
- Both Esmeralda and Bobby had played the game.
- Esmeralda had played the game.
- Only Bobby had played the game.
- Exactly one of the students had played the game.
- Bobby had not played the game.
How does the two-way frequency table assist you in finding the solutions to the above probabilities?
Part II. Determine the probability of the following events.
- Bobby did not play the game given that Esmeralda played it.
- Esmeralda played the game given that Bobby played it.
Part III.
What are the similarities and differences between the events in Part I and Part II?
Discussion/Further Questions/Extensions
Have students make a Venn diagram to pictorially show the relationship given in the frequency table.
Sample Solutions
Part I. Determine the probability that if a game was selected at random from the list—
- P(Both Esmeralda and Bobby had played the game) = 6/100 = 0.06
- P(Esmeralda had played the game) = 10/100 = 0.1
- P(Only Bobby had played the game) = P(Bobby had played the game and Esmeralda had not played the game) = 14/100 = 0.14
- P(Exactly one of the students had played the game) = P(Only Bobby had played the game or only Esmeralda had played the game) = 14/100 + 4/100 = 0.18
- P(Bobby had not played the game) = 80/100 = 0.8
How does the two-way frequency table assist you in finding the solutions to the above probabilities?
Answers may vary: The table organizes the data so that students can more easily see the relationships among the probabilities.
Part II. Determine the probability of the following events.
- P(Bobby did not play the game given that Esmeralda played it) = 4/10 = 0.4
- P(Esmeralda played the game given that Bobby played it) = 6/20 = 0.3
Part III.
What are the similarities and differences between the events in Part I and Part II?
All probabilities can be represented as fractions. In Part I, all denominators are equal to 100, whereas in Part II, the denominators are a subset of the original events.