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.

Deal or No Deal Game show Choose 1 of 26 cases containing $0.01 - $1 million Extremely popular game show on NBC hosted by Howie Mandel A contestant chooses 1 of 26 numbered cases and keeps that case (dollar amounts range from 1 cent to 1 million dollars.) The contestant then opens a certain number of cases he did not open to see what dollar amounts he didn’t win After seeing these dollar amounts that he did not win, he may either open some more cases or take an offer from “the banker” to buy back his case. If this offer is accepted, the contestant sells back his case and takes the offered amount of money and the game is over.

The Game Board .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Enter the 26 dollar amounts into your calculator and compute the mean and the median dollar amounts to win. On the TI-84, click on the STAT button (2nd row from the top in the middle). EDIT should be highlighted with 1:Edit highlighted. Click ENTER. In List L1, enter the values. Type in .01 ENTER in L1(1), 1 ENTER in L1(2), …1000000 ENTER in L1(26).

The Game Board Again Mean: $131477 (x with a bar on top) Median: $875 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $131477 (x with a bar on top) Median: $875 (Med) Once all 26 values are entered into the calculator, please check that the values are correct. One method to validate is to check the stats. Click on STAT and move the curser to the right “>” and CALC is highlighted. Click ENTER and “1-Var Stats will be displayed”. Click ENTER again and the stats will be displayed.

The Game Begins .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 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. We now need to take $10000 off of our game board. Go to STAT and EDIT click ENTER. Scroll down to L1(16)=10000 and click DEL. This will delete this value from our list.

Game Board at 1st Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 After the first 6 cases are opened, click again on STAT, go > to highlight CALC and click ENTER on 1-Var Stats and click enter again to find the stats on the remaining values

Game Board at 1st Deal Mean: $95405 Median: $875 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $95405 Median: $875 What do you think the 1st offer from the bank should be? Let’s get a show of hands as to who: Would sell back their case for $11000 Would say “no deal” Don’t forget that once you sell your case, the game is over and if you don’t sell your case, you may not get an offer as high as this one in the next bank offer The 1st offer from the bank is $11000.

Game Board at 2nd Deal Data from 1st Deal: Mean= $95405 Median = $875 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Data from 1st Deal: Mean= $95405 Median = $875 The first offer from the bank was $11000. What do you think will happen to the offer from the bank? How many values that he eliminated were above the mean value of around $95000 after the first offer and how many were below the mean value. What do you think will happen to the mean of his remaining values?

Game Board at 2nd Deal Mean: $112097 Median: $1000 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $112097 Median: $1000 The 2nd offer from the bank is $24000. Which appears to fluctuate more, the mean or the median? Last time the bank offered $11,000 when the mean was $95405 and the median was $875. What do you think the offer will be now? Would you sell back the case at this stage of the game…if you already sold it before for $11000, you’re out of the game 

Game Board at 3rd Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000

Game Board at 3rd Deal Data from 2nd Deal: Mean= $112097 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Data from 2nd Deal: Mean= $112097 Median = $1000 Bank’s offers: 1st Deal $11000 2nd Deal $24000 Looking at the game board, what do you notice as a pattern…perhaps some gaps developing. How many values did he remove in this round above last round’s mean of $112000. How many were above? What do you think will happen to the banker’s offer?

Game Board at 3rd Deal Mean: $139195 Median: $1000 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $139195 Median: $1000 The 3rd offer from the bank is $50000. Looking at the game board, what do you notice as a pattern…perhaps some gaps developing. How many values did he remove in this round above last round’s mean of $112000. How many were above? What do you think will happen to the banker’s offer? Any ideas on the bank’s offer? How many of you (who haven’t already sold back your case would sell it back now?) Are the deals getting more or less “fair”?

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? Why are the deals getting “better?” Player performance (he’s knocking out more small numbers than larger ones) NBC wants players to go through to the end for higher suspense/ratings and hence they don’t want everyone taking these early deals NBC is trying to minimize their risk of losing a “big value” perhaps any value over a certain threshold ($300,000 and above for example.) Any other ideas?

Game Board at 4th Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 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? Hence, do you think the new offer will be higher or lower?

Game Board at 4th Deal Mean: $153266 Median: $537.50 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 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 4th offer from the bank is $62000. Ideas on Bank’s Offer? Find the percentage of the mean value.

Game Board at 4th Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Two more cases must be eliminated. How many ways can the contestant select two cases? Ask. Wait. Invite participants to explain how they arrived at their number (even if it is wrong).

Game Board at 5th Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Ideas on Bank’s Offer? Find the percentage of the mean value.

Observations Now the mathematics should be visible and intuitive with each case removed. Previous percentages of means have been 12, 21, 36, 41% of the mean…do you think this pattern will continue now that he had a bad round?

Game Board at 5th Deal Mean: $66855 Median: $62.50 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $66855 Median: $62.50 The 5th offer from the bank is $33000. Despite the mean being reduced by more than half, the new offer is 49% of the new mean? What does this say about our conjectures on fairness as the game continues. Who would keep going? Who would take this offer and stop? What has changed now (looks at the board…do you notice patterns such as a gap.)

Game Board at 6th Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Now, what do you think the bank will offer will be? Who would stop now? Keep going?

Game Board at 6th Deal Mean: $80226 Median: $75 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $80226 Median: $75 The 6th offer from the bank is $68000. Now, what do you think the bank will offer will be? Who would stop now? Keep going?

Game Board at 7th Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Now, what do you think the bank will offer will be? Who would stop now? Keep going?

Game Board at 7th Deal Mean: $100033 Median: $62.50 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $100033 Median: $62.50 The 7th offer from the bank is $90000. Last time the offer jumped to $68000, nearly 85% of the mean value. How about now? Any thoughts? Which of you would quit now? Is there anyone left in the audience who is still going? The offer jumped to $90000, nearly 90% of the mean value of the remaining amounts, up from $68000, nearly 85% of the mean value of the remaining amounts before. Even if he eliminates the smallest amount and receives an offer of 100% of the mean, how much will the next offer be? Is the risk worth the reward?

Some Thoughts The offer jumped to $90000, nearly 90% of the mean value of the remaining amounts, up from $68000, nearly 85% of the mean value of the remaining amounts before. Even if he eliminates the smallest amount and receives an offer of 100% of the mean, how much will the next offer be? Is the risk worth the reward?

Game Board at 8th Deal .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000

Game Board at 8th Deal Mean: $133360 Median: $75 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Mean: $133360 Median: $75 The 8th offer from the bank is $137000. What do you think of the offer? What has changed in terms of fairness?

Fairness Crosses the 100% mark. Went from being unfair against the player to fair to unfair in the player’s favor.

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 What do you think of the contestant’s decisions? Intuitively? Mathematically? Did he follow the correct mathematical steps? Do you think you could follow along mathematically and make logical conclusions despite the pressure of the cameras and lack of calculators/computers? Question: Does it really matter what dollar amount was in the case that the contestant chose at the beginning of the game? Is contestant A, who chose a case with $100000 in it guaranteed to win more money on this show than a contestant B who chose a case with $300 in it.

New Game Board 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? .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 2 000 000 During a special presentation of “Deal or No Deal,” the top prize value was changed from $1 million to $2 million while the other 25 values remained the same

New Game Board Original Mean = $131477 New Mean = $169937 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 100 000 100 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 2 000 000 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%. During a special presentation of “Deal or No Deal,” the top prize value was changed from $1 million to $2 million while the other 25 values remained the same Changing 1/26 (or 4%) of the values changes the mean by 29%. Why? Which measure of center is more robust (resistant to change due to extreme values?)

Premier Game Board .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 200 000 100 400 000 200 600 000 300 800 000 400 1 000 000 500 1 500 000 750 2 000 000 NBC doubled all 7 top values during the premier episode. How did this change the mean? How did this change the median?

Premier Game Board Original Mean = $131477 New Mean = $256477 .01 1 000 1 5 000 5 10 000 10 25 000 25 50 000 50 75 000 75 200 000 100 400 000 200 600 000 300 800 000 400 1 000 000 500 1 500 000 750 2 000 000 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%. What if the 7 smallest values were doubled, not the 7 largest values? How would the mean have changed?

Alternative Game Board .02 1 000 2 5 000 10 10 000 20 25 000 50 50 000 100 75 000 150 100 000 200 000 200 300 000 300 400 000 400 500 000 500 750 000 750 1 000 000 Original Mean = $131477 New Mean = $131483 New Mean is a $6 increase over the Old Mean Old Median = $875 New Median = $875 Why did the mean change so much before but not change now?