What is the chance of rolling a 7 on two weighted die?

Question

Say one die is weighted to land on $6$ and the other is weighted to land on the $1$, and a die is four times more likely to land on the weighted side compared to a non-weighted side. What is the probability of the $2$ die summing to seven?

Thought Process: The probability of a die landing on the weighted side is $4/9$. So the probability of both dice landing on the weighted side is $16/81$.

What I'm struggling with is the fact that there is still a chance for the dice to land on some other side, and still sum up to seven. I don't know how to find the total probability.

Answer 1

With the weighted die, the probability is $\frac{4}{9}$ of landing on the weighted side, and $\frac{1}{9}$ of landing on any of the other 5 sides. With two dice, there are 6 combinations that add to 7. (I've put the weighted sides in bold.)

• (1, 6) = (1/9)(1/9) = 1/81
• (2, 5) = (1/9)(1/9) = 1/81
• (3, 4) = (1/9)(1/9) = 1/81
• (4, 3) = (1/9)(1/9) = 1/81
• (5, 2) = (1/9)(1/9) = 1/81
• (6, 1) = (4/9)(4/9) = 16/81

So the total probability is $\frac{16}{81} + 5(\frac{1}{81}) = \frac{21}{81} = \frac{7}{27}$.