1. SO YOU’RE SAYING THERE’S A CHANCE…
Probability in Quantitative Reasoning

2. General QR Info Why QR? How QR? Why probability? How probability?
College Algebra does not seem to fit the bill as a core level math class.
Quantitative Reasoning offers more usable skills and realistic problems
How QR?
Find the realism in the topic by relating it to their experience
Why probability?
We are always making decisions based on the unknown
How probability?
Let’s see…

3. standards
QR.P.1: Determine the nature and number of elements in a finite sample space to model the outcomes of real-world events using counting techniques, and build the sample space by making lists, tables, or tree diagrams.
QR.P.2: Determine the number of ways an event may occur using the Fundamental Counting Principle.
QR.P.3: Evaluate the validity of claims based on empirical, theoretical, and subjective probabilities. Draw conclusions or make decisions related to risk, pay-off, expected value, and false negatives/positives in various probabilistic contexts.
QR.P.4: Use data displays and models, such as two-way tables, tree diagrams, Venn diagrams, and area models, to determine probabilities (including conditional probabilities) and use these probabilities to make informed decisions.

4. standards
QR.P.1: Determine the nature and number of elements in a finite sample space to model the outcomes of real-world events using counting techniques, and build the sample space by making lists, tables, or tree diagrams.
QR.P.2: Determine the number of ways an event may occur using the Fundamental Counting Principle.
QR.P.3: Evaluate the validity of claims based on empirical, theoretical, and subjective probabilities. Draw conclusions or make decisions related to risk, pay-off, expected value, and false negatives/positives in various probabilistic contexts.
QR.P.4: Use data displays and models, such as two-way tables, tree diagrams, Venn diagrams, and area models, to determine probabilities (including conditional probabilities) and use these probabilities to make informed decisions.

5. Using their language Expected Value Risk vs. Reward
Law of Large Numbers
Whatever can happen, will
Conditional Probability
The More You Know…

6. Expected Value Gives a weighted average value for an experiment
EV = P1V1 + P2V2 + … +PnVn
Each Pi is the probability of an event where their union partitions the sample space.

7. Expected Value
Consider a decision that is made based on one piece of information
The decision combined with the outcome contains a risk and reward
Make it a game to start

8. Weather or not…

9. Outcome values 7 10 4

10. Game Play
Monday
Tuesday
Wednesday
Thursday
Friday
50%
30%
70%
20%
40%

11. Continuing with EV Begin to introduce vocabulary
Discuss general strategies
Discuss limitations of expected value
Reveal the existence of an optimal strategy

12. Law of large numbers
One perceived weakness of expected value comes from results-oriented analysis
To counteract this issue, you can discuss repeating the experiment numerous times to show that you will get optimal results eventually
This leads to discussing issues that are addressed by the Law Of Large Numbers

13. And now it’s…

14. LLN and data clumping
Excel is a great tool to simulate multiple trials
Large numbers of trials will produce data clumping
Experiencing this from different perspectives can lead to seemingly paradoxical situations

15. Coin Flips
Use the rand(), randbetween(), countif(), and if() Excel functions, as well as conditional formatting to generate large numbers of trials and interpret the results
Example: Flip a coin times and find the longest streak(s)

16. Phone Insurance
A phone insurance company offer $500 to replace a broken phone and its customers have a 1% chance of breakage each month
What is the probability that a customer does not break their phone for a year?
What is the probability that a customer breaks their phone exactly once in a year?

17. Phone Insurance Use Excel to simulate 1000 customers over 12 months
Record the number of breaks per month, per year, and double breaks in a year
Check that these fit in with your earlier results

18. Conditional Probability
Sometimes we receive new information about the outcome of an experiment
The new information restricts the sample space thus changing the probability calculation
This is often a good transition to statistical testing

19. Cooler Problem
Three families (J,K,L) each bring identical coolers to a cookout. Cooler J contains half Coke, half Pepsi. Cooler K has ¾ Coke, ¼ Pepsi. Cooler L is all Coke. You reach into one cooler and grab a drink.
What is the probability you chose cooler K?
What is the probability you get a Pepsi?
If you get a Coke, what is the probability you chose cooler J?

20. Tree diagram
2 12 J K L ⅓
2 12
3 12
1 12 1
4 12

21. J K L Tree diagram ½ What is the probability you chose cooler K? ½ ⅓ ¾
2 12 J K L ⅓
What is the probability you chose cooler K?
2 12
3 12
1 12 1
4 12

22. J K L Tree diagram ½ What is the probability you get a Pepsi? ½ ⅓ ¾ ¼
2 12 J K L ⅓
What is the probability you get a Pepsi?
2 12
3 12
1 12 1
4 12

23. Tree diagram
2 12 J K L ⅓
If you get a Coke, what is the probability you chose cooler J?
2 12
3 12
1 12 1
4 12

24. Multiple representations
While a tree diagram is “standard”, other representations may be more useful in understanding conditional probability

25. Table 2 12 2 12 3 12 1 12 4 12 ½ ½ ⅓ ⅓ ¾ ¼ ⅓ ⅓ ⅓ ⅓ Coke Pepsi Cooler J
Cooler K
Cooler L
2 12
2 12 ⅓ ⅓
3 12
1 12 ⅓ ⅓ 1
4 12 ⅓ ⅓

26. Area model J K L

27. Thanks!