Game Show Math

[object Object],Bowen Kerins Senior Curriculum Designer, EDC (and part-time game consultant) [email_address]

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[object Object],Want to win? We’ll need some volunteers for three games . You could win as much as, um, $4 in prizes!

CME Project ,[object Object],[object Object],[object Object],[object Object],[object Object]

CME Project Overview ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object]

[object Object],February 2000 (for $1000: How many degrees in a right angle?)

[object Object],February 2000 (I got the next question wrong.)

[object Object],2004 (Double overbid on the showcase! Bummer.)

[object Object],Summer 2007 (I also worked on “Show Me The Money” and “The Singing Bee”… which, amazingly, is still on the air.)

[object Object],A popular show! Better known as “ Deal or No Deal” $.01 $1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $750 $1,000 $5,000 $10,000 $25,000 $50,000 $75,000 $100,000 $200,000 $300,000 $400,000 $500,000 $750,000 $1,000,000

[object Object],The “ fair deal ”: Multiply each outcome by its probability… Total: $410,210 Fair deal: ~$102,500 $.01 $1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $750 $1,000 $5,000 $10,000 $25,000 $50,000 $75,000 $100,000 $200,000 $300,000 $400,000 $500,000 $750,000 $1,000,000

[object Object],The “ bank offer ”: Guarantee, almost always less than fair value Fair deal: ~$102,500 Offer: $82,000 Deal or No Deal ? $.01 $1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $750 $1,000 $5,000 $10,000 $25,000 $50,000 $75,000 $100,000 $200,000 $300,000 $400,000 $500,000 $750,000 $1,000,000

[object Object],$.01 $1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $750 $1,000 $5,000 $10,000 $25,000 $50,000 $75,000 $100,000 $200,000 $300,000 $400,000 $500,000 $750,000 $1,000,000 What’s the expected value of the initial board?

[object Object],$.01 $1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $750 $1,000 $5,000 $10,000 $25,000 $50,000 $75,000 $100,000 $200,000 $300,000 $400,000 $500,000 $750,000 $1,000,000 Initial board… Fair deal: $131,477 First offer: ~$15,000-$20,000 Why do the first offers stink?

[object Object],$.01 $1 $5 $10 $25 $50 $75 $100 $200 $300 $400 $500 $750 $1,000 $5,000 $10,000 $25,000 $50,000 $75,000 $100,000 $200,000 $300,000 $400,000 $500,000 $750,000 $1,000,000 Winnings per player: ~$122,500 Offer percentages (compared to fair value, by round): 11%, 21%, 36%, 50%, 62%, 73%, 88%, 92%, 98%

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[object Object],Place a price next to each item. If it’s the right price, you win the prize! Almond Joy 8-Pack Cracker Jack 3-Pack Gold Bond Powder Juicy Fruit 2-Pack

[object Object],Place a price next to each item. If it’s the right price, you win the prize! Almond Joy 8-Pack Cracker Jack 3-Pack Gold Bond Powder Juicy Fruit 2-Pack 96¢ 97¢ 98¢ $1.00 Good luck… you’ll need it. Audience, any advice?

[object Object],So, how did you do…? Almond Joy 8-Pack $1.00 Cracker Jack 3-Pack 98¢ Gold Bond Powder 97¢ Juicy Fruit 2-Pack 96¢

[object Object],If I keep offering this game repeatedly, how many prizes will I give away, on average ? Calculate the expected value for the number of prizes per game. There are 24 different ways the player can place the price tags. (Why?)

[object Object],Here are the 24 ways to place the price tags: ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA

[object Object],For each of the 24 ways, count the number of price tags placed correctly. ABCD 4 AB DC 2 A CB D 2 A CDB 1 A DBC 1 A D C B 2 BA CD 2 BADC 0 BCA D 1 BCDA 0 BDAC 0 BD C A 1 CAB D 1 CADB 0 C B A D 2 C B DA 1 CDAB 0 CDBA 0 DABC 0 DA C B 1 D B AC 1 D BC A 2 DCAB 0 DCBA 0

[object Object],This frequency chart shows the number of ways to get each result. 24 TOTAL 9 0 8 1 6 2 1 4 # ways # prizes

[object Object],This frequency chart shows the number of ways to get each result. The total number of prizes is 4 x 1 + 2 x 6 + 1 x 8 … 24 prizes and 24 ways. Divide to find… Hey, it’s 1! 24 TOTAL 9 0 8 1 6 2 1 4 # ways # prizes

[object Object],Reconsider the problem from Jack’s perspective. (Cracker Jack, that is.) ABCD ABDC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB CBAD CBDA CDAB CDBA DABC DACB DBAC DBCA DCAB DCBA

[object Object],Light up all the places where prize B is correctly placed. A B CD A B DC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB C B AD C B DA CDAB CDBA DABC DACB D B AC D B CA DCAB DCBA

[object Object],Prize B is won one-fourth (6 out of 24) of the time… and all the prizes are like that. A B CD A B DC ACBD ACDB ADBC ADCB BACD BADC BCAD BCDA BDAC BDCA CABD CADB C B AD C B DA CDAB CDBA DABC DACB D B AC D B CA DCAB DCBA

[object Object],With four prizes, each is won 1/4 of the time. 4 x (1/4) = 1 With five prizes, each is won 1/5 of the time. With n prizes, each is won 1/ n of the time. n x (1/ n ) = 1 Clever methods can “beat” writing it all out.

The mean (average) given away is 1. But what about the standard deviation ? Calculate the variance , the “mean squared deviation”, then take its square root. 24 deviations: value - mean . What are they? A follow-up question

[object Object],For each of the 24 ways, determine the deviation, which is the score minus 1. ABCD 3 AB DC 1 A CB D 1 A CDB 0 A DBC 0 A D C B 1 BA CD 1 BADC -1 BCA D 0 BCDA -1 BDAC -1 BD C A 0 CAB D 0 CADB -1 C B A D 1 C B DA 0 CDAB -1 CDBA -1 DABC -1 DA C B 0 D B AC 0 D BC A 1 DCAB -1 DCBA -1

[object Object],For each of the 24 ways, determine the deviation… then square them. ABCD 9 AB DC 1 A CB D 1 A CDB 0 A DBC 0 A D C B 1 BA CD 1 BADC 1 BCA D 0 BCDA 1 BDAC 1 BD C A 0 CAB D 0 CADB 1 C B A D 1 C B DA 0 CDAB 1 CDBA 1 DABC 1 DA C B 0 D B AC 0 D BC A 1 DCAB 1 DCBA 1

The variance is… This frequency chart shows the number of ways to get each squared deviation. 24 TOTAL 8 0 15 1 1 9 # ways sq. dev.

The variance is… This frequency chart shows the number of ways to get each squared deviation. The total squared deviation is 9 x 1 + 1 x 15 + 0 x 8 = 24 from 24 values. Divide… OMG 1 again! 24 TOTAL 8 0 15 1 1 9 # ways sq. dev.

[object Object],PLINKO! Sadly, you can’t play Plinko without its board. Plinko is played so often that great data is available: 2000-2008: played 258 times! Total chips: 1,056 Total chips in $10,000 space: 152 ( 14.4% ) Total winnings: $1,908,600 ( $1,807 per chip) Average winnings per play : $7,398

[object Object],Where you drop Plinko chips matters a lot ! $100 $500 $1,000 $0 $10,000 Chip EV Drop Above

[object Object],Where you drop Plinko chips matters a lot ! $780 $100 $1,009 $500 $1,606 $1,000 $2,266 $0 $2,558 $10,000 Chip EV Drop Above

[object Object],Where you drop Plinko chips matters a lot ! The table says… DROP IT IN THE MIDDLE!!! Actual average winnings: $1,807 per chip (guess why…) $780 $100 $1,009 $500 $1,606 $1,000 $2,266 $0 $2,558 $10,000 Chip EV Drop Above

[object Object],ANY NUMBER! That’s much easier to play! Who wants to play a game??

[object Object],7 4 6 3 8 5 1 0 9 CAR RANGE PIGGY BANK

[object Object],Assuming the player is just picking randomly (which seems about right) , what is the probability that they win the car? This is a hard question!

[object Object],Assuming the player is just picking randomly, what is the probability that they win the car? This question would be a lot easier if the car had 3 digits instead of 4… Why must the probability of winning the car be less than 1/3?

[object Object],There are 10! = 3,628,800 different ways the player can pick numbers. 10! is a much bigger number than 24, so enumerating by hand is impractical. One option is to simulate running the game a large number of times. Here’s 10,000 trials: Car: 2,605 (26.05%) Range: 3,682 Piggy Bank: 3,713

[object Object],Solve a simpler version! If there’s one number in each prize left, there is a 1/3 chance of winning the car. If there’s two numbers left in the car and one in each of the small prizes, there is a 2/4 x 1/3 = 1/6 chance of winning the car. This is like a coordinate system: P(2,1,1) = 1/6 . Continue and “build out” until you find the answer at P(4,3,3) . (Build a 3-D model!)

[object Object],For computers, 3.6 million isn’t big, it’s around the number of 5-card poker hands. A computer can try all 10! ways the game could be played: This is different than the simulation found; its probability is an estimate. Car: 933,120 (25.71% = 9/35) Range: 1,347,840 Piggy Bank: 1,347,840

[object Object],Let’s play a different game called Any Number But That One . You pick a number; if it’s in a prize, that prize explodes . You win the last prize standing. Say you picked 2… the car explodes! (Oops.) What’s the probability of winning this game?

[object Object],This game doesn’t last nearly as long… On the first pick, 6 of the 10 numbers explode a “bad” prize. After you blow up one bad prize, there’s one more decisive pick: 3 of the 7 numbers explode the other “bad” prize to win the car. 6/10 x 3/7 = 9/35 (25.71%) Hey, it’s the same probability…

[object Object],Players win Any Number more often than predicted by chance. 2000-2008 194 plays Big prize ( A New Car! ) : 70 (36.08%) Little prize: 65 Piggy Bank: 59 Players often guess the first digit of the car. Also, 0 and 5 are more likely to appear there.

[object Object],In my teaching, I found some game shows worked better than others. Mostly I used games for test review, but also for openers or wrap-ups. Good Press Your Luck Card Sharks Millionaire High Rollers Bad Jeopardy! (yes) Deal or No Deal Twenty-One Sale of the Century

[object Object],NBN was The Price is Right with bingo instead of shopping. (It only lasted six episodes. Wonder why…) Its games were interesting probability problems, but usually no strategy. Except for Bingo 500… Who wants to play?

[object Object],The final game of the night is Bingo 500 . If you complete a bingo before the contestant wins, you win all remaining prizes! (Ties will be broken by a math problem.) Important: if you are one number away from a bingo, STAND UP so we can tell you are one away.

[object Object],For each bingo ball, the contestant guesses whether the next ball will be higher or lower than the one that just came out. If they are right, the ball number is added to their score. 75 is better than 23. The first ball is a “freebie”. The contestant wins if they get a total of 500 points or more before anyone in the audience completes a bingo. (The audience: 200 players, one card each.)

[object Object],Is the next ball higher or lower than 42? Bingo balls go from 1 to 75 . 42

[object Object],Is the next ball higher or lower than 60? 42 60

[object Object],Is the next ball higher or lower than 5? 42 60 5

[object Object],Is the next ball higher or lower than 30? Remember: if you are one away, STAND UP . 42 60 5 30

[object Object],Is the next ball higher or lower than 46? 42 60 5 30 46

[object Object],Is the next ball higher or lower than 24? 42 60 5 30 46 24

[object Object],Is the next ball higher or lower than 1? (Don’t think too hard about this one.) 42 60 5 30 46 24 1

[object Object],Is the next ball higher or lower than 47? Remember: if you are one away, STAND UP . 42 60 5 30 46 24 1 47

[object Object],Is the next ball higher or lower than 64? 42 60 5 30 46 24 1 47 64

[object Object],Is the next ball higher or lower than 39? (What would you do here? This one is kind of in the middle.) 42 60 5 30 46 24 1 47 64 39

[object Object],Is the next ball higher or lower than 72? 42 60 5 30 46 24 1 47 64 39 72

[object Object],Is the next ball higher or lower than 56? Getting closer…? 42 60 5 30 46 24 1 47 64 39 72 56

[object Object],Is the next ball higher or lower than 67? 42 60 5 30 46 24 1 47 64 39 72 56 67

[object Object],Is the next ball higher or lower than 41? 42 60 5 30 46 24 1 47 64 39 72 56 67 41

[object Object],Is the next ball higher or lower than 18? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18

[object Object],Is the next ball higher or lower than 3? Bad time for such a low number! 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3

[object Object],Is the next ball higher or lower than 48? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48

[object Object],Is the next ball higher or lower than 71? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71

[object Object],Is the next ball higher or lower than 26? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71 26

[object Object],Is the next ball higher or lower than 53? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71 26 53

[object Object],Is the next ball higher or lower than 28? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71 26 53 28

[object Object],Is the next ball higher or lower than 31? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71 26 53 28 31

[object Object],Is the next ball higher or lower than 14? 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71 26 53 28 31 14

[object Object],(That’s all we got, but someone should win by now…!) 42 60 5 30 46 24 1 47 64 39 72 56 67 41 18 3 48 71 26 53 28 31 14 66

[object Object],Did strategy change as the game went on? One strategy is to go higher if in the bottom half, otherwise lower . Is this the best strategy? Or what? What’s this got to do with summations? Balance? Similar triangles?? Players did not follow good strategy on the actual show: the first contestant to play Bingo 500 picked “lower” on a 30. (He lost.)

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[object Object],[object Object],Bowen Kerins Senior Curriculum Designer, EDC (and part-time game consultant) [email_address]

Game Show Math

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