Question

A game of chance offers the following odds and payoffs:

Probability         Payoff

0.2                      $500

0.4                        100

0.4                 0

a) What is the expected cash payoff?

b) Suppose each play of the game costs $100. What is the expected rate of return?

c) What is the variance of the expected returns?

d) What is the standard deviation of the expected returns?

Answer 1

a)

Expected cash payoffs = Sum of product of payoffs and respective probablities

Probablity (a)	Payoff(b)	a*b
0.2	500	100
0.4	100	40
0.4	0	0
	Expected payoff =	= $140

b)

rate of return = (profit/cost) * 100

Profit = payoff -cost

Expected rate of return = Sum of product of rate of returns and respective probablities

Probablity (a)	Profit	Rate of return (b)	a*b
0.2	= 500-100 = 400	= (400/100)*100 = 400%	=80%
0.4	= 100-100 = 0	= (0/100) * 100 = 0	=0
0.4	= 0-100 = -100	= (-100/100) * 100 = -100%	=-40%
		Expected rate of return =	=40%

c)

Probablity (a)	Rate of return (r)	(r-R)^2 (b)	a*b
0.2	400%	=54442.89	=10888.58
0.4	0	=27783.89	=11113.56
0.4	100%	=4444,89	=1777.96
		Sum of deviations=	=23780.1

R= Mean of rate of returns = (400+0+100) / 3 = 166.67

Variance = sum of deviation / number of outcomes = 23780.1 / 3 = 7926.69

d)

standard deviation = square root of variance

=7926.69 = 89.03%