A carnival operator is constructing a new game In this game

A carnival operator is constructing a new game.

In this game, the player will place a bet. Then a ball is selecteded from an urn containing balls of three colours: 2 green balls, 4 red balls and 7 white balls.

If a red ball is drawn, the player wins 3$, if a green ball is drawn, the player wins 4$, and if a white ball is drawn the player wins nothing.

What should the carnival operator set the bet at such that on average he makes $0.50 profit a game? (That is, so the player loses $0.50)

Solution

P(g) = 2/13

P(r) = 4/13

P(w) = 6/13

Avg. winnng to a player without including bet money = P(g) * money from green + P(r) money from red + P(w) * money from white = 2/13*4 + 4/13 *3 + 7/13 * 0=1.538 $

So, bet = earning + 0.5 $ = 1.538 + 0.5 = 2.038 $ per bet