In a bowling "frame", between 0 and 10 pins might be knocked over by a bowling ball. Suppose that a particular bowling alley attracts a mixture of beginners and experts, such that we can assume that the probability mass function (pmf) of the number of pins knocked over per frame is roughly symmetric. Further, suppose that the pmf has a mean of 5.7 pins and standard deviation of 1.9 pins, and that the results of all frames are independent.
A person attends the bowling alley one evening, and bowls 31 frames. What would be the probability that they knocked over more than 200 pins?