Question

A friend offered to play a game with you where you select 1 marble out of 100: 40 red 30 blue 20 yellow and 10 green. If you pick a red marble you lose $1, if you pick a blue you win $2, if you pick a yellow you lose $5 if you pick a green you win $10. Each time you pick a marble it will go back into the bag. Round to 2 decimals and show work.

a) What is the expected value of this game? Interpret the meaning of the expected value.

b) If you were able to play this game 500 times, how many times do you expect to win?

c) Find and interpret the variance and the standard deviation of the game.

d) If each original outcome value decreased by 1 dollar, what would the new expected value, variance and standard deviation be?

e) If each original outcome value was multiplied by 4 what would the new expected value, variance, and standard deviation be?

Answer 1

Game details:
Total balls , n = 100    , Red balls = 40 , Blue balls = 30 , Yellow balls = 20 , Green balls = 10
Selected ball	Total balls	Win amount in $ (x)	P(X=x)	x * P(X=x)	* P(X=x)
Red	40	-1	40/100 = 0.40	-0.40	0.400
Blue	30	2	30/100 = 0.30	0.60	1.200
Yellow	20	-5	20/100 = 0.20	-1.00	5.000
Green	10	10	10/100 = 0.10	1.00	10.000

P(X=Red) = Total Red Balls/ Total available balls = 40 / 100 = 040 . Same way probability is computed for other color balls in above table.

a.

Expected value of game , E(x) = ,

Expected value indicates that possibility of winning in each game is 0.20.

b) If you were able to play this game 500 times, how many times do you expect to win?

Total available balls = 100 . We win each time when we pick blue or green ball.

Probability of picking Blue ball from above table  P(Blue) = 30/100 = 0.30

Probability of picking Green ball P(Green) =  10/100 = 0.10

Hence , P(Blue Green) = P(Blue) + P(Green) = 0.30 + 0.10 = 0.40 .

Hence , if we play this game 500 times, Probability of winning bets = 500 * P(Blue Green) = 500 * 0.40 = 200 times.

c) Find and interpret the variance and the standard deviation of the game.

from above table , variance =

Standard deviation indicates the spread of values from their mean, while variance indicates the measure of average degree to which each value differs from the average of all data points.

d) If each original outcome value decreased by 1 dollar, what would the new expected value, variance and standard deviation be?

Here would be game details, expected value if each outcome value is decreased by 1 dollar. Now if Red ball is picked, amount lost is $2 , if blue ball is picked , amount win is $1 ....

Game details:
Total balls , n = 100    , Red balls = 40 , Blue balls = 30 , Yellow balls = 20 , Green balls = 10
Selected ball	Total balls	Win amount in $ (x)	P(X=x)	x * P(X=x)	sqr ( x) * P(X=x)
Red	40	-2	40/100 = 0.40	-0.80	1.600
Blue	30	1	30/100 = 0.30	0.30	0.300
Yellow	20	-6	20/100 = 0.20	-1.20	7.200
Green	10	9	10/100 = 0.10	0.90	8.100

Expected value of game , E(x) = ,

from above table , variance =