Question

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?

Answer 1

The probability that you will roll a 7 is 0.14.

Explanation:

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`.