Mathematics

Problem

In a game, a player needs to throw a ball as far as possible. They win the amount of money (in Yen) equal to the length of the throw, in metres. Let

X

be the continuous random variable modelling the length of a throw. It is assumed that no one can throw further than 10 m, and that the probability density function of

X

f(x) = \frac{0.328e^x}{e^x + e^7} \text{ for } 0 \le x \le 10

a. a How much should be charged per game in order to make the game fair? b. b What percentage of players are expected to make a profit from this game?

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