Newton’s Dice

There is a famous (among statisticians) question posed to Isaac Newton about which is more likely:  rolling at least 1 six when 6 dice are rolled, rolling at least 2 sixes when 12 dice are rolled, or at least 3 sixes when 18 dice are rolled.  This is usually used to illustrate the difference between averages and probability.  While you would expect at least 1 six when 6 dice are rolled, 2 when rolling 12 etc. they do not all have the same probability of occurring.  It turns out that the more dice you roll the closer that probability gets to 0.5, a coin toss.  But with smaller numbers of dice you are more likely to roll the average and therefore have a higher probability of having at least the average.

Who cares? you might ask.  Well this bit of trivia came in handy on Saturday.  My in-laws taught me a little game called Liars Dice which is played as follows.  Each player starts with six dice.  Ones are wild.  Players shake the dice in an opaque cup, slam it down and look so that players only see their own dice.  The player starting the round declares how many of a given number, including ones, he or she believes are on the table.  The next person in the line can either raise the number of dice keeping the initial numeral the same or raise the numeral keeping the number of dice the same.  For example, if the first player guesses that there are 7 fours, the next player can say (1) there are 8 fours, or (2) there are 7 fives.  You don’t have to increment by one in either case, but you can’t go lower.  The bidding goes around until someone’s bluff is called and the dice are counted.  If the challenger is right and there are fewer dice than claimed then the bidder loses one die and the game repeats.  If the challenger is wrong, he or she loses one.

The point is I lost…big time.  Being able to write out the explicit probability formula of the entire game was nothing but a cheap parlour trick that amused my father-in-law.  Knowing the probabilities is not enough to win, especially if you thrive off the thrill of pushing the odds.  That must be why Vegas continues to attract so many people even during a recession.  It’s about the thrill, because if it was about the odds no one would play.  I’m glad I stuck with my original Sin City strategy not to play or I may have been lost the farm.


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