Ralph and Claudia are playing a game in which the higher score wins. Ralph's scores in this game are (approximately) normally distributed with a mean of 100 and a standard deviation of 20. Claudia's scores in this game are (approximately) normally distributed with a mean of 110 and a standard deviation of 15.
a) Who is more likely to score above 150?
b) Assuming their scores are independent (which is not true for many games but is at least approximately true for some other games), what is the (approximate) probability that Ralph beats Claudia?
c) Now suppose they play three games and declare the winner to be the one who gets the highest total for the three games together. What is the (approximate) probability that Ralph beats Claudia in this format?
d) This is a pretty silly problem. But why is the mathematics involved so important? (Think of some examples where we have used this same sort of reasoning to do more useful things.)
e) One more change. This time they play three games but the winner is the one who wins at least two of the three. What is the (approximate) probability that Ralph beats Claudia in this format?