Probability with weights

Question

Assume that the numbers 1 through 4 are equally likely, as are the numbers 5 and 6, yet rolling a 1 is four times as likely as rolling a 5. (1) If we have probability weights w1 through w6, one for each possible outcome, what is the the value of w5?

Answer 1

The prob's of 1 - 4 are four times as much as a 5 or a 6. So we can call each of 1 - 4 a weight of 4, and 5 and 6 a weight of 1 each. With four 4s and 2 1s, this gives us a total weight of 4+4+4+4+1+1 = 18. Divide all the weights by 18, and we see that w5 = 1/18.