Question

A pair of dice is loaded. The probability that a 4 appears on the first die is 2/7, and the

probability that a 3 appears on the second die is 2/7. Other outcomes for each die appear

with probability 1/7. What is the probability of 7 appearing as the sum of the numbers

when the two dice are rolled?

Answer 1

First die

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 1 & 2 & 3 & 4 & 5 & 6\\ \hline p(x) & 1/7 & 1/7 & 1/7 & 2/7 & 1/7 & 1/7\\ \end{array}$$

Second die

$$\def\arraystretch{1.5} \begin{array}{c:c} x & 1 & 2 & 3 & 4 & 5 & 6\\ \hline p(x) & 1/7 & 1/7 & 2/7 & 1/7 & 1/7 & 1/7\\ \end{array}$$

$$1+6=2+5=3+4=4+3=5+2=6+1$$

$$P(sum=7)=\dfrac{1}{7}(\dfrac{1}{7})+\dfrac{1}{7}(\dfrac{1}{7})+\dfrac{1}{7}(\dfrac{1}{7})$$

$$+\dfrac{2}{7}(\dfrac{2}{7})+\dfrac{1}{7}(\dfrac{1}{7})+\dfrac{1}{7}(\dfrac{1}{7})=\dfrac{9}{49}$$