2023 AMC 10B Problems/Problem 19

Solution

Since all the actions are independent, we can switch the orders. Let Sonya choose the direction first. And the problem is symmetric, so we consider just one direction. WLOG, let's say she choose . When she first pick the location, she'll have to be within 1 unit of the axis to have a chance to jump out of the boundary southward. That's . With in that region, the expected y coordinate would be 0.5 which is 0.5 unit from the boundary (x-axis). Now, the jumping distance required to jump out of the boundary on average has to be greater than 0.5. That's another . So the final probability is . ~Technodoggo