Two 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?

Let #x# equal the probability that you will roll a 1. This will be the same probability as rolling a 3, 5, or 6. The probability of rolling a 2 or a 4 is #3x#. We know that these probabilities must add to one, so

The probability of rolling a 1 + the probability of rolling a 2 + the probability of rolling a 3 + the probability of rolling a 4 + the probability of rolling a 5 + the probability of rolling a 6 = 1.

#x+3x+x+3x+x+x=1#

#10x=1#

#x=0.1#

So the probability of rolling a 1, 3, 5, or 6 is 0.1 and the probability of rolling a 2 or a 4 is #3(0.1)=0.3#.

There are a limited number of ways of rolling the dice to have the sum shown on the dice to equal to 7.

First die = 1 (probability 0.1)
Second die = 6 (probability 0.1)

Probability of this happening is #(0.1)(0.1)=0.01#

First die = 2 (probability 0.3)
Second die = 5 (probability 0.1)

Probability of this happening is #(0.3)(0.1)=0.03#

First die = 3 (probability 0.1)
Second die = 4 (probability 0.3)

Probability of this happening is #(0.1)(0.3)=0.03#

First die = 4 (probability 0.3)
Second die = 3 (probability 0.1)

Probability of this happening is #(0.3)(0.1)=0.03#

First die = 5 (probability 0.1)
Second die = 2 (probability 0.3)

Probability of this happening is #(0.1)(0.3)=0.03#

First die = 1 (probability 0.1)
Second die = 6 (probability 0.1)

Probability of this happening is #(0.1)(0.1)=0.01#

Now we can sum all of these probabilities

Probability of rolling a 7 is

#0.01+0.03+0.03+0.03+0.03+0.01=0.14#.