Home > Probability
| Dice Game (Posted on 2002-07-24) |
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I have set up a stall where you may play a fabulous game.
Presented before you are 4 unusual 6-sided dice:
A Big Red die:
has the numbers 5, 5, 5, 5, 1 and 1 on its sides
A Large Yellow die:
has the numbers 6, 6, 2, 2, 2 and 2 on its sides
A Medium Green die:
has the numbers 6, 4, 4, 2, 2 and 1 on its sides
A Small Blue die:
has the numbers 3, 3, 3, 3, 3 and 1 on its sides
I inform those that are unaware that the average value they would roll with each of the 4 dice are (roughly) 3.66, 3.33, 3.17 and 2.67 respectively. All dice are fair and players find it impossible to cheat when rolling them.
I request a $1 payment from you to play. You may choose any one die. Then I may choose any of the remaining dice. We then roll. If you roll more than or the same as me, I return your original $1 stake and a bonus $1 prize. If I score more than you, I keep your stake and you win nothing.
What would be your strategy if you wanted to walk away from my stall with the most amount of money possible?
(Thanks go to an old university professor would showed us something similar, which instantly intrigued me, in a Probability lecture)
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Submitted by Nick Reed
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Rating: 4.0000 (9 votes)
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Solution:
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Although the game seems geared towards the player winning, it simply isn't so.
Misconception 1: I will pick the big red dice - it has the highest average, so I'm more likely to win
Misconception 2: I win if I roll higher OR on a draw - winning on a draw gives me that extra edge
Misconception 3: I either lose $1 or gain $1 - combined with the above I'm sure to make money in the long run!
If the person running the stall knows the basic premise they will, on average, win more often than not:
If the player picks the red die, the stall-owner should pick the yellow
If the player picks the yellow die, the stall-owner should pick the blue
If the player picks the green die, the stall-owner should pick the red
If the player picks the blue die, the stall-owner should pick the green
If the stall-owner does this, regardless of which die the player picks, the stall-owner has a 20/36 chance (i.e. 55.55%) of winning due to the specific way the dice are numbered. The player's chances of winning are less than 50% - as the payout is only what the player would normally lose, the stall-owner will make money in the long run.
So, in reply to the question: "The only winning move is not to play"
(Also noteworthy, as noticed by TomM, is that Red beats Blue 24/36 times - so if forced to play, the best strategy is not to pick Blue - however, the best strategy is still not to play at all) |
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