Question

14. Imagine you are in a game show. There are 10 prizes hidden on a game board with 100 spaces. One prize is worth $50, three are worth $20, and another six are worth $10. You have to pay $5 to the host if your choice is not correct. Let the random variable x be the winning.

a. Complete the following probability distribution. (Show the probability in fraction format and explain your work)

x

P(x)

-$5

$10

$20

$50

b. What is your expected winning or loss in this game? Be specific in your answer whether it’s winning or loss. (Show your work and round the answer to two decimal places)

c. What is the standard deviation of the probability distribution? (Show your work and round the answer to two decimal places)

Answer 1

Since, total trials are 100 and 10 are prizes, then,

probabilities can be calculated by dividing frequency by total trials , that is 100

x	-$5	$10	$20	$50
Frequency	90	6	3	1
P(x)	0.9	0.06	0.03	0.01

Expected[ Win/Loss ] = ( 6*10 ) + ( 50*1 ) + ( 20*3 ) - ( 5*100 ) , as one always have to pay to play4

Expected[ Win/Loss ] = 60 + 50 + 60 - 500 = 170 - 500 = - 330,

this is always a condition of loss, since the expected value is -330

mean of x = Expected[ Win/Loss ]/100 = -3.3

Variance = ( -5*-5*90 + 10*10*6 + 20*20*3 + 50*50*1 )/100 - Expected[ Win/Loss ]^2

Variance = ( 2250 + 600 + 1200 + 2500 )/100 - (-330*-330)

Variance = 6550/100 - 10.89 = 65.5 - 10.89 = 54.61

Standard deviation = 54.61^0.5 = 7.39