Suppose you are given a coin with a 0 on one side and a 30 on the other. You als

Question

Suppose you are given a coin with a 0 on one side and a 30 on the other. You also have a standard six sided die with values one through six.

• If you were to flip the coin 10 times and roll the die 10 times and record the data as variables f and r, which would likely have the highest mean
outcome? Which would likely have the highest variance?

Here I assume that the coin would have the largest mean outcome and variance, as the mean would be 15 x10 with the coin compared to 3.5 x 10 with the dice. With the variance of the coin we would have 152 x 2 vs all the sides of the dice minus the mean squared (coin would still be much higher) over 9. Is this a reasonable explaination? Am I incorrect?

• Suppose a casino starts a game where the contestant rolls the die, flips the coin, and is paid the total of the two outcomes. (For example, rolling a 2 and having the coin come up 30 pays $32.) What is the minimum amount the casino must charge to make a profit on the game. (We are assuming the casino has no costs other than the payout.)

In order for the Casino to make a profit they would need to charge at least the sum of the two means correct? So at least 18.5 to play?

Explanation / Answer

let X denotes the outcome of the coin.

and Y denotes the outcome of the dice.

then X=0 with probability 0.5

=30 with probability 0.5

so mean of X=E[X]=0*0.5+30*0.5=15

the probability distribution of Y is

Y: 1 2 3 4 5 6

P[Y=y]= 1/6 1/6 1/6 1/6 1/6 1/6

so mean of Y is E[Y]=(1+2+3+4+5+6)/6=3.5

variance of X is =V[X]=E[X2]-E2[X]

now E[X2]=0*0*0.5+30*30*0.5=450

so V[X]=450-152=450-225=225

variance of Y is V[Y]=E[Y2]-E2[Y]

now E[Y2]=(1*1+2*2+3*3+4*4+5*5+6*6)/6=91/6

so V[Y]=91/6-3.52=35/12

now the coin is flipped 10 times and the dice is rolled 10 times.

the data are stored as variables f and r

so f=10X and r=10Y

so mean of f is E[f]=10*E[X]=10*15=150

mean of r is E[r]=10*E[Y]=10*3.5=35

hence mean of f is highest [answer]

variance of f is V[f]=100*V[X]=100*225=22500

variance of r is V[r]=100*V[Y]=100*35/12=291.66

so the variance of coin is also highest.. [answer]

the amount paid to a customer is a random variable which is the sum of two outcomes.

so if Z denotes the amount the customer gets then Z=X+Y

so the minimum amount the casino must charge to make a profit on the game. (assuming the casino has no costs other than the payout.) is E[Z]=E[X]+E[Y]=15+3.5=$18.5 [answer]

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