They are all the same, but if you take a number's reciprocal and square it by the number, and the number is a power of 2, you get the square root of 2, NOT 1. This is because when you simplify the other equations, you get the square root of 2, every single time under these conditions. Remember, fractions as exponents are square roots to the power of the reciprocal of the fraction. Therefore they are all the same. Around 1.41421356237 ish, but the decimal never ends.
First calculate the probability that the larger number is n (1 <= n <= 6). This is exactly (2n-1)/36. Why? There are three possibilities:
The first die is exactly n, the second die is smaller -- So second die can be 1 ... n-1.
The second die is exactly n, the first die is smaller -- So the first die can be 1 ... n-1.
Both die are the same. This happens in only one way.
So total number of ways in which the larger number is n = 2n-1. Hence the probability that the larger number is n is (2n - 1)/36.
Consequently, the expected value of the larger of the two numbers is
Sum [n = 1 to 6] (2n -1)/36 * n
= 1/36 (Sum [n = 1 to 6] 2n^2 - Sum [n = 1 to 6] n)
You can compute this either directly or by using formulas for sequences. The answer is 161/36.
Since the expected loss is 50 cents, we know it is more likely to lose than to win. It is also impossible to get -50, and the total difference between the 2 possibilities is 2, 50/200=1/4, so the chance of winning is 1/4 and the chance of losing is 3/4. Since (1/4)*3=3/4, the amount of black balls are 3 times the amount of white balls, so k=15.
Hope this helps!! :)