Originally Posted by
indio
All we do to get the proper numbers is to look at every scenario and figure out the chances. This is done with a binomial calculator.
There are 121 scenarios possible when Andy and Bobby take 10 free throws each. Andy will make 10,9,8,7,6,5,4,3,2,1,or 0, and Bobby will make 10,9,8,7,6,5,4,3,2,1,or 0. This gives us 121 possibilities each with separate probabilities of occurring. For example, we can figure the chances of a draw occurring by adding together the chances of each draw result, of which there are 11. Some of the draw results have such a ridiculously low chance of happening that we can ignore them. the chances of 0-0,1-1, and 2-2, are off the charts (a 2-2 score has about a 1.8 million to one chance). The draw score lines that we add up would be 4-4, 5-5, 6-6, 7-7, 8-8, 9-9, and 10-10.
For example, Andy has a 28.16% chance of making exactly 8 free throws, and Bobby has a 23.35% chance of making exactly 8 free throws, so obviously the chances that the final score being 8-8 would be .2816 x .2335 for a total of 6.58%. When we add up all the draw chances, we get our 19.27% chance of a draw.
4-4 = 0.06
5-5 = 0.60
6-6 = 2.92
7-7 = 6.68
8-8 = 6.58
9-9 = 2.27
10-10 = 0.16
draw = 19.27%
And you can get the picture of how we can evaluate the 55 ways Andy can beat Bobby, and the 55 ways Bobby can beat Andy..