Loaded Dice

Two “loaded” dice each have the property that a 2 or a 4 is three times as likely to appear as a 1,3,5, or 6 on each roll. What is the probability that a 7 will be the sum when the two dice are rolled?
Source: NCTM Mathematics Teacher, February 2006

SOLUTION
If the dice are not loaded, each number 1,2,3,\cdots\!,6 is equally likely to come up. The probability of rolling each number on a single die is 1/6. Note that 1/6+1/6+1/6+1/6+1/6+1/6=1.
What if the dice are loaded in favor of 2 and 4?
If the probability of rolling a 1,3,5, or 6 equals 1/x, the probability of rolling a 2 or 4 equals 3/x.
1/x+1/x+1/x+1/x+3/x+3/x=1
10/x=1
x=1/10
The probability of rolling a 1,3,5, or 6 on a single die equals 1/10 and the probability of rolling a 2 or 4 on a single die equals 3/10.
The sum of 7 occurs when two dice are rolled as follows
1,6
2,5
3,4
4,3
5,2
6,1
The probability of a sum of 7
\dfrac{1}{10}\cdot\dfrac{1}{10}+\dfrac{3}{10}\cdot\dfrac{1}{10}+\dfrac{1}{10}\cdot\dfrac{3}{10}+\dfrac{3}{10}\cdot\dfrac{1}{10}+\dfrac{1}{10}\cdot\dfrac{3}{10}+\dfrac{1}{10}\cdot\dfrac{1}{10}=\dfrac{14}{100}=\dfrac{7}{50}

Answer: 7/50

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About mvtrinh

Retired high school math teacher.
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