Basic Probability, part 3

August 23, 2009

Dice01

Dice are very old instruments for games, thus they had been found in Egyptian tombs, dated to 2000 BC. Dice gamble were greatly popular in ancient Greek, Rome, Asia, then in medieval Europe and it has been being popular today. In ancient times the throw of dice was believed to be controlled by the gods. Famous was the game with throwing two dice and guessing the sum of the numbers. People noticed that the number 7 as the sum turns out more frequently and they thought that the cause is the gods favour. Number 7 is the lucky number!

Today with maths help we can understand why the sum 7 appears more frequently.

Consider all possible cases in rolling two dice. The first die has 6 sides and 6 numbers, the same is for the second die – 6 sides with 6 numbers. All possible sums of these numbers are from 1+1=2 to 6+6=12.

The picture below presents the table with all possible cases, each intersection presents the sum of the numbers of two dice:
15

As you can see the sum 7 appears more frequently than any other sum. There are 6 cases with the sum 7 and there are 36 possible cases at all. Hence the probability that the sum of two rolled dice will be 7 is \frac{6}{36}=\frac{1}{6}.

The probability for the sum 7 is the greatest. For example, for the sum 5 the probability is \frac{4}{36}=\frac{1}{9}, since there are only four cases we can get the sum 5.

Moreover, the probability of getting the sum 7 is equal to the probability of getting the sum (2 or 3 or 11 or 12), since P(Sum ~2 ~or~ 3~ or~ 11~ or~ 12)= \frac{1}{36}+\frac{2}{36}+\frac{2}{36}+\frac{1}{36}=\frac{6}{36}=\frac{1}{6}.

Ancient people used the number 7 of the sum two dice to check cheating. If the sum 7 appeared often then dice are fair, if not then dice are loaded. Brilliant method!

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