Question

Consider a lottery game in which a person can win $0, $1, $2, or $1,000. The probability of winning nothing when one plays the game is 0.99, the probability of winning $1 is 0.009, and the probability of winning $2 is 0.0009.

1. If the game cost $1 to play what is the probability that a person will at least win their money back in the game?
2. What is the interval of values that are within one standard deviation of the mean?
3. What is the probability that the lottery winnings will be within one standard deviation of the mean?

Answer 1

X	P(X)	X*P(X)	X² * P(X)
0	0.99	0.0000	0.0000
1	0.009	0.0090	0.0090
2	0.0009	0.0018	0.0036
1000	0.0001	0.1000	100.0000

	P(X)	X*P(X)	X² * P(X)
total sum =	1	0.1108	100.01

a)

probability that a person will at least win their money back in the game = P(X≥1) = 0.009+0.0009+0.0001 =0.01

b)

mean = E[X] = Σx*P(X) =            0.1108
          
E [ X² ] = ΣX² * P(X) =            100.0126
          
variance = E[ X² ] - (E[ X ])² =            100.0003
          
std dev = √(variance) =            10.0000

interval of values that are within one standard deviation of the mean = (0.1108±1*10) = (-9.89 , 10.11 )

c)

probability that the lottery winnings will be within one standard deviation of the mean = P(-9.89<x< 10.11 ) = P(X=0)+P(X=1) + P(X=2) = 0.9999