Max has a spinner that lands on 1 with a probability of $\frac{1}{2}$, lands on 2 with a probability of $\frac{1}{4}$, lands on 3 with a probability of $\frac{1}{6}$, and lands on 4 with a probability of $\frac{1}{12}$. if max spins the spinner, and then zack spins the spinner, then what is the probability that max gets a larger number than zack does?

If Max lands on 4, then Zack can land on 3, 2, 1. The probability that Zack lands on 3, 2, 1 is 1/6 + 1/4 + 1/2 = 11/12.

The combined probability of Max landing on 4 and Zack landing on 3, 2, 1 is:
1/12 * 11/12 = 11/144.

If Max lands on 3, then Zack can land on 2, 1. The probability that Zack lands on 2, 1 is 1/4 + 1/2 = 3/4.

The combined probability of Max landing on 3 and Zack landing on 2, 1 is:
1/6 * 3/4 = 1/8.

If Max lands on 2, then Zack must land on 1. The probability of this is 1/4 * 1/2 = 1/8.

If Max lands on 1, then he cannot get a larger number than Zack.

The overall probability that Max has a larger number than Zack is:
11/144 + 1/8 + 1/8 = 47/144