Price is Right "Punchboard": Probabilities
June 29th, 2018 at 7:00:09 PM
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Hi all.
I'm curious to work out the expected value of one game of Price is Right's "Punchboard". Here are the rules.
The player gets up to four selections from the punchboard. For the sake of this math exercise, let's assume the player always gets four selections.
The player looks at their first selection, and must decide whether to keep it, or throw it back and look at their next selection. If they get down to selection #4, they automatically get whatever it is. Their selections do NOT accumulate.
The distribution is as follows:
$25,000 - 1
$10,000 - 2
$5,000 - 4
$2,500 - 8
$1,000 - 10
$500 - 10
$250 - 10
$100 - 5
Total - 50
For the sake of this exercise, let's assume the following:
1) The player will always have four selections.
2) The player will always walk away with $2,500 or higher, and will always continue playing with $1,000 or lower.
Based on these assumptions, the player will walk away on their first turn 30% of the time, and will remain in the game 70% of the time. From there, however, I'm lost.
What do you think? Thanks in advance.
I'm curious to work out the expected value of one game of Price is Right's "Punchboard". Here are the rules.
The player gets up to four selections from the punchboard. For the sake of this math exercise, let's assume the player always gets four selections.
The player looks at their first selection, and must decide whether to keep it, or throw it back and look at their next selection. If they get down to selection #4, they automatically get whatever it is. Their selections do NOT accumulate.
The distribution is as follows:
$25,000 - 1
$10,000 - 2
$5,000 - 4
$2,500 - 8
$1,000 - 10
$500 - 10
$250 - 10
$100 - 5
Total - 50
For the sake of this exercise, let's assume the following:
1) The player will always have four selections.
2) The player will always walk away with $2,500 or higher, and will always continue playing with $1,000 or lower.
Based on these assumptions, the player will walk away on their first turn 30% of the time, and will remain in the game 70% of the time. From there, however, I'm lost.
What do you think? Thanks in advance.
Casinos are not your friends, they want your money. But so does Disneyland.
And there is no chance in hell that you will go to Disneyland and come back with more money than you went with.
- AxelWolf and Mickeycrimm
June 29th, 2018 at 9:22:55 PM
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Quote: DeucekiesHi all.
I'm curious to work out the expected value of one game of Price is Right's "Punchboard". Here are the rules.
The player gets up to four selections from the punchboard. For the sake of this math exercise, let's assume the player always gets four selections.
The player looks at their first selection, and must decide whether to keep it, or throw it back and look at their next selection. If they get down to selection #4, they automatically get whatever it is. Their selections do NOT accumulate.
The distribution is as follows:
$25,000 - 1
$10,000 - 2
$5,000 - 4
$2,500 - 8
$1,000 - 10
$500 - 10
$250 - 10
$100 - 5
Total - 50
For the sake of this exercise, let's assume the following:
1) The player will always have four selections.
2) The player will always walk away with $2,500 or higher, and will always continue playing with $1,000 or lower.
Based on these assumptions, the player will walk away on their first turn 30% of the time, and will remain in the game 70% of the time. From there, however, I'm lost.
What do you think? Thanks in advance.
I’ve watched this game and I don’t think the player generally walks away with $2500 if that’s their first punch and they have three remaining
June 30th, 2018 at 4:14:28 AM
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Quote: DeucekiesHi all.
I'm curious to work out the expected value of one game of Price is Right's "Punchboard". Here are the rules.
The player gets up to four selections from the punchboard. For the sake of this math exercise, let's assume the player always gets four selections.
The player looks at their first selection, and must decide whether to keep it, or throw it back and look at their next selection. If they get down to selection #4, they automatically get whatever it is. Their selections do NOT accumulate.
The distribution is as follows:
$25,000 - 1
$10,000 - 2
$5,000 - 4
$2,500 - 8
$1,000 - 10
$500 - 10
$250 - 10
$100 - 5
Total - 50
For the sake of this exercise, let's assume the following:
1) The player will always have four selections.
2) The player will always walk away with $2,500 or higher, and will always continue playing with $1,000 or lower.
Based on these assumptions, the player will walk away on their first turn 30% of the time, and will remain in the game 70% of the time. From there, however, I'm lost.
What do you think? Thanks in advance.
He ends up on the low side:
(35/50) * (34/49) * (33/48) * (32/47) = 0.2273556231
Ending up on the highside would be 1 - 0.2273...
Find the average of the high side and average of the low side. Weight them, obviously (ie: you have four 5000's but eight 2500's). Then multiply the average by the likelihood to end up on that side. And that's your EV.
June 30th, 2018 at 5:14:15 AM
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Quote: michael99000I’ve watched this game and I don’t think the player generally walks away with $2500 if that’s their first punch and they have three remaining
That would be the SMART thing to do. If your first punch is 2500, keep it! The rest could be $100 for all you know!
In both The Hunger Games and in gambling, may the odds be ever in your favor. :D
"Man Babes" #AxelFabulous
"Olive oil is processed but it only has one
ingredient, olive oil."-Even Bob, March 27/28th. :D The 2 year war is over! Woo-hoo! :D
I sometimes speak in metaphors. ;) Remember this. ;)
Crack the code. :D 8.9.13.25.14.1.13.5.9.19.14.1.20.8.1.14! :D
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June 30th, 2018 at 5:26:01 AM
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My non-math guy thoughts.
EV on a free roll is always positive. So I'm not sure that's the term for what you're asking. But if, for this exercise, $2500+ is a win, and anything less is a loss, maybe it could be expressed that way.
Total value of the game is $98000. 50 choices means average prize value is $1960. So I think you have the right stop point identified, and the risk of going on past a $2500 draw is a question of human nature more than mathematics.
Your chance of getting at least $2500 on one of four punches seems nearly certain.
First punch =15/50 or 30.00%.
2nd Punch = 15/49 or 30.60%.
Third punch = 15/48 or 31.25%.
4th punch = 15/47 or 31.91%.
So your chance improves with each failure. But you still have a better than 2/3 chance of failure on each individual turn.
Conversely, you should win slightly less than 1/3 of the time each punch, but you get 4 chances. While that doesn't guarantee you'll find one of the 15 winners in 4 punches, you should be able to add the 4 winning percentages together, and the amount over 100 should be accurate.
Adding them, i get 123.77%, or a 23.77% edge.
Using the same method, but going on $5000 or better, with 7 slips, I get.
14.00%
14.29%
14.59%
14.89%
For a chance of 57.77%, or a bit better than half the time, you will get $5000 or more. And still on that 4th punch, you have some chance of hitting one of the remaining $2500's for a better than average return. There could be 5,6,7,or 8 of those left when you make your 4th punch.
Figuring out the odds of the remaining $2500s is a bit beyond me, so I'll SWAG it's about 28% added to the 57. Call it 85% you'll end up a winner (as defined by above average).
Be aware, this could be a totally wrong way to evaluate it. But it's fun to try. Thanks for the question!
Edit. RS, who's better at this than I am, posted while I was figuring. I would take what he said over mine. I'm leaving mine up anyway.
EV on a free roll is always positive. So I'm not sure that's the term for what you're asking. But if, for this exercise, $2500+ is a win, and anything less is a loss, maybe it could be expressed that way.
Total value of the game is $98000. 50 choices means average prize value is $1960. So I think you have the right stop point identified, and the risk of going on past a $2500 draw is a question of human nature more than mathematics.
Your chance of getting at least $2500 on one of four punches seems nearly certain.
First punch =15/50 or 30.00%.
2nd Punch = 15/49 or 30.60%.
Third punch = 15/48 or 31.25%.
4th punch = 15/47 or 31.91%.
So your chance improves with each failure. But you still have a better than 2/3 chance of failure on each individual turn.
Conversely, you should win slightly less than 1/3 of the time each punch, but you get 4 chances. While that doesn't guarantee you'll find one of the 15 winners in 4 punches, you should be able to add the 4 winning percentages together, and the amount over 100 should be accurate.
Adding them, i get 123.77%, or a 23.77% edge.
Using the same method, but going on $5000 or better, with 7 slips, I get.
14.00%
14.29%
14.59%
14.89%
For a chance of 57.77%, or a bit better than half the time, you will get $5000 or more. And still on that 4th punch, you have some chance of hitting one of the remaining $2500's for a better than average return. There could be 5,6,7,or 8 of those left when you make your 4th punch.
Figuring out the odds of the remaining $2500s is a bit beyond me, so I'll SWAG it's about 28% added to the 57. Call it 85% you'll end up a winner (as defined by above average).
Be aware, this could be a totally wrong way to evaluate it. But it's fun to try. Thanks for the question!
Edit. RS, who's better at this than I am, posted while I was figuring. I would take what he said over mine. I'm leaving mine up anyway.
Last edited by: beachbumbabs on Jun 30, 2018
If the House lost every hand, they wouldn't deal the game.
June 30th, 2018 at 9:39:14 AM
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With one punch left, the expected value is $2060.
With two punches left, stand on $2500 or more.
With three punches left, stand on $5000 or more.
With four punches left, stand on $5000 or more.
With two punches left, stand on $2500 or more.
With three punches left, stand on $5000 or more.
With four punches left, stand on $5000 or more.
"For with much wisdom comes much sorrow." -- Ecclesiastes 1:18 (NIV)
June 30th, 2018 at 10:32:52 AM
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I find for the best expected value, shoot for $5,000 or more. You have a 22.56% chance of success with EV of $4,785.73.
Regarding the OP question of settling for $2,500 or more, you have a 77.28% chance of success with an EV of $4,495.79.
Aim one prize lower for $1,000 or better and you have a 94.5% chance of success and EV of $3,607.03.
Target prize: 25000
Success probability: 0.07988189999999995
Final choice probability: 1-0.0199843 2-0.0199962 3-0.0200472 4-0.9399723
Prize 25000 probability 0.0798819 expected value 1997.0475000000001
Prize 10000 probability 0.0374662 expected value 374.662
Prize 5000 probability 0.0749742 expected value 374.87100000000004
Prize 2500 probability 0.150257 expected value 375.6425
Prize 1000 probability 0.1879456 expected value 187.94559999999998
Prize 500 probability 0.1877103 expected value 93.85515
Prize 250 probability 0.1876929 expected value 46.923225
Prize 100 probability 0.0940719 expected value 9.40719
Overall expected value: 3460.3541649999997
Target prize: 10000
Success probability: 0.22557839999999996
Final choice probability: 1-0.0599802 2-0.0575175 3-0.055209 4-0.8272933
Prize 25000 probability 0.0752063 expected value 1880.1575
Prize 10000 probability 0.1503721 expected value 1503.721
Prize 5000 probability 0.0659012 expected value 329.506
Prize 2500 probability 0.1317927 expected value 329.48175000000003
Prize 1000 probability 0.1647656 expected value 164.7656
Prize 500 probability 0.1648364 expected value 82.4182
Prize 250 probability 0.1646732 expected value 41.168299999999995
Prize 100 probability 0.0824525 expected value 8.24525
Overall expected value: 4339.4636
Target prize: 5000
Success probability: 0.4641223
Final choice probability: 1-0.1401739 2-0.122749 3-0.1074715 4-0.6296056
Prize 25000 probability 0.0664128 expected value 1660.32
Prize 10000 probability 0.1326851 expected value 1326.851
Prize 5000 probability 0.2650244 expected value 1325.122
Prize 2500 probability 0.0996456 expected value 249.114
Prize 1000 probability 0.124617 expected value 124.617
Prize 500 probability 0.1246395 expected value 62.31975
Prize 250 probability 0.1246427 expected value 31.160674999999998
Prize 100 probability 0.0623329 expected value 6.233289999999999
Overall expected value: 4785.737715
Target prize: 2500
Success probability: 0.7727732
Final choice probability: 1-0.3000898 2-0.2141886 3-0.1519695 4-0.3337521
Prize 25000 probability 0.0514948 expected value 1287.3700000000001
Prize 10000 probability 0.1031071 expected value 1031.071
Prize 5000 probability 0.2059888 expected value 1029.944
Prize 2500 probability 0.4121825 expected value 1030.45625
Prize 1000 probability 0.0650534 expected value 65.0534
Prize 500 probability 0.0648724 expected value 32.4362
Prize 250 probability 0.064867 expected value 16.216749999999998
Prize 100 probability 0.032434 expected value 3.2434
Overall expected value: 4495.790999999999
Target prize: 1000
Success probability: 0.9449961
Final choice probability: 1-0.5001046 2-0.2549386 3-0.127566 4-0.1173908
Prize 25000 probability 0.0376984 expected value 942.46
Prize 10000 probability 0.0756081 expected value 756.081
Prize 5000 probability 0.1513757 expected value 756.8785
Prize 2500 probability 0.3024555 expected value 756.13875
Prize 1000 probability 0.3778584 expected value 377.85839999999996
Prize 500 probability 0.022033 expected value 11.0165
Prize 250 probability 0.0219736 expected value 5.4934
Prize 100 probability 0.0109973 expected value 1.0997299999999999
Overall expected value: 3607.02628
Target prize: 500
Success probability: 0.9940398
Final choice probability: 1-0.7000239 2-0.214353 3-0.062409 4-0.0232141
Prize 25000 probability 0.0284108 expected value 710.27
Prize 10000 probability 0.0568556 expected value 568.556
Prize 5000 probability 0.1137848 expected value 568.924
Prize 2500 probability 0.226955 expected value 567.3874999999999
Prize 1000 probability 0.2838669 expected value 283.8669
Prize 500 probability 0.2841667 expected value 142.08335
Prize 250 probability 0.0039947 expected value 0.9986750000000001
Prize 100 probability 0.0019655 expected value 0.19654999999999997
Overall expected value: 2842.2829749999996
Target prize: 250
Success probability: 0.999979
Final choice probability: 1-0.9000122 2-0.091833 3-0.0076404 4-0.0005144
Prize 25000 probability 0.0222559 expected value 556.3974999999999
Prize 10000 probability 0.0444259 expected value 444.25899999999996
Prize 5000 probability 0.0888231 expected value 444.1155
Prize 2500 probability 0.177862 expected value 444.655
Prize 1000 probability 0.2223524 expected value 222.35240000000002
Prize 500 probability 0.2221854 expected value 111.09270000000001
Prize 250 probability 0.2220743 expected value 55.518575
Prize 100 probability 0.000021 expected value 0.0021
Overall expected value: 2278.3927750000003
Target prize: 100
Success probability: 1
Final choice probability: 1-1 2-0 3-0 4-0
Prize 25000 probability 0.0200208 expected value 500.52
Prize 10000 probability 0.0399329 expected value 399.329
Prize 5000 probability 0.0798928 expected value 399.464
Prize 2500 probability 0.1600258 expected value 400.0645
Prize 1000 probability 0.2002462 expected value 200.24620000000002
Prize 500 probability 0.1999822 expected value 99.9911
Prize 250 probability 0.1997429 expected value 49.935725
Prize 100 probability 0.1001564 expected value 10.015640000000001
Overall expected value: 2059.5661649999997
PM me if you want the Javascript code. Too damn many brackets to deal with to post with a spoiler.
Regarding the OP question of settling for $2,500 or more, you have a 77.28% chance of success with an EV of $4,495.79.
Aim one prize lower for $1,000 or better and you have a 94.5% chance of success and EV of $3,607.03.
Target prize: 25000
Success probability: 0.07988189999999995
Final choice probability: 1-0.0199843 2-0.0199962 3-0.0200472 4-0.9399723
Prize 25000 probability 0.0798819 expected value 1997.0475000000001
Prize 10000 probability 0.0374662 expected value 374.662
Prize 5000 probability 0.0749742 expected value 374.87100000000004
Prize 2500 probability 0.150257 expected value 375.6425
Prize 1000 probability 0.1879456 expected value 187.94559999999998
Prize 500 probability 0.1877103 expected value 93.85515
Prize 250 probability 0.1876929 expected value 46.923225
Prize 100 probability 0.0940719 expected value 9.40719
Overall expected value: 3460.3541649999997
Target prize: 10000
Success probability: 0.22557839999999996
Final choice probability: 1-0.0599802 2-0.0575175 3-0.055209 4-0.8272933
Prize 25000 probability 0.0752063 expected value 1880.1575
Prize 10000 probability 0.1503721 expected value 1503.721
Prize 5000 probability 0.0659012 expected value 329.506
Prize 2500 probability 0.1317927 expected value 329.48175000000003
Prize 1000 probability 0.1647656 expected value 164.7656
Prize 500 probability 0.1648364 expected value 82.4182
Prize 250 probability 0.1646732 expected value 41.168299999999995
Prize 100 probability 0.0824525 expected value 8.24525
Overall expected value: 4339.4636
Target prize: 5000
Success probability: 0.4641223
Final choice probability: 1-0.1401739 2-0.122749 3-0.1074715 4-0.6296056
Prize 25000 probability 0.0664128 expected value 1660.32
Prize 10000 probability 0.1326851 expected value 1326.851
Prize 5000 probability 0.2650244 expected value 1325.122
Prize 2500 probability 0.0996456 expected value 249.114
Prize 1000 probability 0.124617 expected value 124.617
Prize 500 probability 0.1246395 expected value 62.31975
Prize 250 probability 0.1246427 expected value 31.160674999999998
Prize 100 probability 0.0623329 expected value 6.233289999999999
Overall expected value: 4785.737715
Target prize: 2500
Success probability: 0.7727732
Final choice probability: 1-0.3000898 2-0.2141886 3-0.1519695 4-0.3337521
Prize 25000 probability 0.0514948 expected value 1287.3700000000001
Prize 10000 probability 0.1031071 expected value 1031.071
Prize 5000 probability 0.2059888 expected value 1029.944
Prize 2500 probability 0.4121825 expected value 1030.45625
Prize 1000 probability 0.0650534 expected value 65.0534
Prize 500 probability 0.0648724 expected value 32.4362
Prize 250 probability 0.064867 expected value 16.216749999999998
Prize 100 probability 0.032434 expected value 3.2434
Overall expected value: 4495.790999999999
Target prize: 1000
Success probability: 0.9449961
Final choice probability: 1-0.5001046 2-0.2549386 3-0.127566 4-0.1173908
Prize 25000 probability 0.0376984 expected value 942.46
Prize 10000 probability 0.0756081 expected value 756.081
Prize 5000 probability 0.1513757 expected value 756.8785
Prize 2500 probability 0.3024555 expected value 756.13875
Prize 1000 probability 0.3778584 expected value 377.85839999999996
Prize 500 probability 0.022033 expected value 11.0165
Prize 250 probability 0.0219736 expected value 5.4934
Prize 100 probability 0.0109973 expected value 1.0997299999999999
Overall expected value: 3607.02628
Target prize: 500
Success probability: 0.9940398
Final choice probability: 1-0.7000239 2-0.214353 3-0.062409 4-0.0232141
Prize 25000 probability 0.0284108 expected value 710.27
Prize 10000 probability 0.0568556 expected value 568.556
Prize 5000 probability 0.1137848 expected value 568.924
Prize 2500 probability 0.226955 expected value 567.3874999999999
Prize 1000 probability 0.2838669 expected value 283.8669
Prize 500 probability 0.2841667 expected value 142.08335
Prize 250 probability 0.0039947 expected value 0.9986750000000001
Prize 100 probability 0.0019655 expected value 0.19654999999999997
Overall expected value: 2842.2829749999996
Target prize: 250
Success probability: 0.999979
Final choice probability: 1-0.9000122 2-0.091833 3-0.0076404 4-0.0005144
Prize 25000 probability 0.0222559 expected value 556.3974999999999
Prize 10000 probability 0.0444259 expected value 444.25899999999996
Prize 5000 probability 0.0888231 expected value 444.1155
Prize 2500 probability 0.177862 expected value 444.655
Prize 1000 probability 0.2223524 expected value 222.35240000000002
Prize 500 probability 0.2221854 expected value 111.09270000000001
Prize 250 probability 0.2220743 expected value 55.518575
Prize 100 probability 0.000021 expected value 0.0021
Overall expected value: 2278.3927750000003
Target prize: 100
Success probability: 1
Final choice probability: 1-1 2-0 3-0 4-0
Prize 25000 probability 0.0200208 expected value 500.52
Prize 10000 probability 0.0399329 expected value 399.329
Prize 5000 probability 0.0798928 expected value 399.464
Prize 2500 probability 0.1600258 expected value 400.0645
Prize 1000 probability 0.2002462 expected value 200.24620000000002
Prize 500 probability 0.1999822 expected value 99.9911
Prize 250 probability 0.1997429 expected value 49.935725
Prize 100 probability 0.1001564 expected value 10.015640000000001
Overall expected value: 2059.5661649999997
PM me if you want the Javascript code. Too damn many brackets to deal with to post with a spoiler.
“You don’t bring a bone saw to a negotiation.” - Robert Jordan, former U.S. ambassador to Saudi Arabia
June 30th, 2018 at 10:43:46 AM
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I don't get how this can be arrived at without considering what choices have already been seen.Quote: WizardWith one punch left, the expected value is $2060.
With two punches left, stand on $2500 or more.
With three punches left, stand on $5000 or more.
With four punches left, stand on $5000 or more.
“You don’t bring a bone saw to a negotiation.” - Robert Jordan, former U.S. ambassador to Saudi Arabia
