1. Deal or No Deal? Fair or Not Fair? Dr. Jason Gershman Rice University School Mathematics Project This presentation is property of Jason Gershman and the Rice University School Mathematics Project. This project is in no way affiliated with NBC/Universal, Endemol USA, “Deal or No Deal” or any of its affiliated companies or sponsors.

2. Deal or No Deal Game show Choose 1 of 26 cases containing $0.01 - $1 million

3. The Game Board.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

4. The Game Board Again.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $131477 (x with a bar on top) Median: $875 (Med)

5. The Game Begins First, the contestant chooses his case and this is put aside, sealed. He then opens cases that he didn’t choose. The first case he opens contains $10000..011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

6. Game Board at 1 st Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

7. Game Board at 1 st Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 The 1 st offer from the bank is $11000. Mean: $95405 Median: $875 What do you think the 1 st offer from the bank should be?

8. Game Board at 2 nd Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Data from 1 st Deal: Mean= $95405 Median = $875 The first offer from the bank was $11000.

9. Game Board at 2 nd Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $112097 Median: $1000 The 2 nd offer from the bank is $24000.

10. Game Board at 3 rd Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

11. Game Board at 3 rd Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Data from 2 nd Deal: Mean= $112097 Median = $1000 Bank’s offers: 1 st Deal $11000 2 nd Deal $24000

12. Game Board at 3 rd Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $139195 Median: $1000 The 3 rd offer from the bank is $50000.

13. Fairness Offer : Mean Ratio First Deal : 11000/95405 = 12% of Mean Second Deal: 24000/112097= 21% of Mean Third Deal: 50000/139195 = 36% of Mean Why are the deals so “unfair” Why do you think NBC is willing to pay a greater percentage of the mean now than before?

14. Game Board at 4th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 What is the average of the 3 amounts that he removed ($300000, $5000 and $25?) Is this above or below the previous mean value of $139195?

15. Game Board at 4th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $153266 Median: $537.50 What percentage of the mean value (12%, 21%, 36 % previously in order) do you think the offer will be? The 4 th offer from the bank is $62000.

16. Game Board at 4th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Two more cases must be eliminated. How many ways can the contestant select two cases?

17. Game Board at 5th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

18. Observations

19. Game Board at 5th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $66855 Median: $62.50 The 5 th offer from the bank is $33000.

20. Game Board at 6th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

21. Game Board at 6th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $80226 Median: $75 The 6 th offer from the bank is $68000.

22. Game Board at 7th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

23. Game Board at 7th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $100033 Median: $62.50 The 7 th offer from the bank is $90000.

24. Some Thoughts

25. Game Board at 8th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000

26. Game Board at 8th Deal.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Mean: $133360 Median: $75 The 8 th offer from the bank is $137000.

27. Fairness

28. Conclusions What do you think of the contestant’s decisions? Does it really matter which case the contestant chooses at the start of the game? Comments / Questions

29. New Game Board.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7502 000 000 The top prize changes from $1 million to $2 million while the other 25 values remain the same. How does this change the mean? The median? Does changing this one value affect the expected outcome in the game?

30. New Game Board.011 000 15 000 510 000 1025 000 2550 000 5075 000 75100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7502 000 000 Original Mean = $131477 New Mean = $169937 –New Mean is a 29% increase over the Old Mean Old Median =$875 New Median=$875 –Changing 4% of the values changes the median by 0%.

31. Premier Game Board.011 000 15 000 510 000 1025 000 2550 000 5075 000 75200 000 100400 000 200600 000 300800 000 4001 000 000 5001 500 000 7502 000 000 NBC doubled all 7 top values during the premier episode. –How did this change the mean? –How did this change the median?

32. Premier Game Board.011 000 15 000 510 000 1025 000 2550 000 5075 000 75200 000 100400 000 200600 000 300800 000 4001 000 000 5001 500 000 7502 000 000 Original Mean = $131477 New Mean = $256477 –New Mean is a 95% increase over the Old Mean Old Median =$875 New Median=$875 –Changing 27% of the values changes the median by 0%.

33. Alternative Game Board.021 000 25 000 1010 000 2025 000 5050 000 10075 000 150100 000 100200 000 200300 000 300400 000 400500 000 500750 000 7501 000 000 Original Mean = $131477 New Mean = $131483 –New Mean is a $6 increase over the Old Mean Old Median =$875 New Median=$875