Deal or No Deal? Fair or Not Fair? Dr. Jason Gershman Rice University School Mathematics Project...
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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.
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Transcript of Deal or No Deal? Fair or Not Fair? Dr. Jason Gershman Rice University School Mathematics Project...
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
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
The Game Board Again.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)
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.
.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 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
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
The 1st offer from the bank is $11000.
• Mean: $95405• Median: $875
What do you think the 1st offer from the bank should be?
Game Board at 2nd 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
Data from 1st Deal:
Mean= $95405Median = $875
The first offer from the bank was $11000.
Game Board at 2nd 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
• Mean: $112097• Median: $1000
The 2nd offer from the bank is $24000.
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.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= $112097Median = $1000
Bank’s offers:1st Deal $110002nd Deal $24000
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
• Mean: $139195• Median: $1000
The 3rd offer from the bank is $50000.
FairnessOffer : 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?
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?
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
• 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.
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?
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
Observations
0
10
20
30
40
50
60
70
80
90
100
110
120
0 1 2 3 4 5 6 7 8
Round
Off
er (
% o
f M
ean
)
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
• Mean: $66855• Median: $62.50
The 5th offer from the bank is $33000.
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
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
• Mean: $80226• Median: $75
The 6th offer from the bank is $68000.
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
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
• Mean: $100033• Median: $62.50
The 7th offer from the bank is $90000.
Some Thoughts
0
10
20
30
40
50
60
70
80
90
100
110
120
0 1 2 3 4 5 6 7 8
Round
Off
er
(% o
f M
ean
)
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.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.
Fairness
0
10
20
30
40
50
60
70
80
90
100
110
120
0 1 2 3 4 5 6 7 8
Round
Off
er (
% o
f M
ean
)
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
New 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 2 000 000
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?
New 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 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%.
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.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%.
Alternative Game Board.02 1 000
2 5 000
10 10 000
20 25 000
50 50 000
100 75 000
150 100 000
100 200 000
200 300 000
300 400 000
400 500 000
500 750 000
750 1 000 000
Original Mean = $131477New Mean = $131483
– New Mean is a $6 increase over the Old Mean
Old Median = $875 New Median = $875
