In a popular casino game, you can bet one whether a ball will fall in an arc on a wheel colored red, black, or green. Say the probability of a red outcome is StartFraction 18/38 EndFraction 18/38 , that of a black outcome is StartFraction 18 / 38 EndFraction 18/38 , and that of a green outcome is StartFraction 2/38 EndFraction 2/38. Suppose someone makes a $20 bet on black. Find the expected net winnings for this single bet.

The expected net winnings for the bet are - $1.0526

Step-by-step explanation:

P (x = + $20) = P (Black outcome) = 18/38

P (x = -$20) = P (red outcome) + P (green outcome)

= 18/38 + 2/38 = 20/38

Hence the probability distribution of x = $20, P (x) = 18/38

x = - $20, P (x) = 20/38

Expected value of the random variable x is given by;

miu = Summation [xP (x) ] = 20 (18/38) - 20 (20/38)

= - $1.0526

hence, the expected net winnings for the bet are - $1.0526

This implies that if a player bet on a very large number of games, the player would on the average lose $1.0526 per single bet