Question

For a particular peculiar pair of dice, the probabilities of rolling $1, 2, 3, 4, 5,$ and $6$ on each die are in the ratio $1 : 2 : 3 : 4 : 5 : 6.$ What is the probability of rolling a total of $7$ on the two dice?

$\dfrac{4}{63}$

$\dfrac{1}{8}$

$\dfrac{8}{63}$

$\dfrac{1}{6}$

$\dfrac{2}{7}$

Answer: C

Concepts:dice (probability)basic probability

Difficulty rating: 1740

Solution:

Since the weights sum to $21,$ the probability of rolling $k$ is $\dfrac{k}{21}.$

A total of $7$ comes from $(1,6), (2,5), \ldots, (6,1),$ so the probability is $\begin{aligned} &\scriptsize \frac{1 \cdot 6 + 2 \cdot 5 + 3 \cdot 4 + 4 \cdot 3 + 5 \cdot 2 + 6 \cdot 1}{21^2} \\ &= \frac{56}{441} = \frac{8}{63}. \end{aligned}$

Thus, the correct answer is C.

2006 AMC 12B Problem 17

Answer: C

Difficulty rating: 1740

Solution:

Problem 17 in Other Years