If you bet $1 in Kentucky’s pick 4 lottery, you either lose $1 or gain $4,999. (

Question

If you bet $1 in Kentucky’s pick 4 lottery, you either lose $1 or gain $4,999. (The winning
prize is $5000, but your $1 bet is not returned. The game is played by selecting a four digit number between 0000 and 9999.

(a) What is the probability of winning?

(b) If you bet $1 on the number 1234 a hundred times, what is the expected value of your total gain/loss across the 100 plays? Show steps. Be sure to write any formulas you are using before plugging into them. Highlight your final answer. (Hint: your gain/loss is the sum of 100 draws from a box)

(c) Calculate the standard error of your total gain/loss across 100 plays. Show steps. Be
sure to write any formulas you are using before plugging into them.

Explanation / Answer

(a) Total number between 0000 and 9999 = 104

Probabiliy of winning : 1/104 = 0.0001

(b) P(win) = 104 and P(loose) = 1-104

E(loss) for 100 plays : 100E(loss)

= 100((0.0001)(4999)+(0.9999)(-1)) = -50

(c) Standard error = standard deviation/n0.5

variance = E(X2)-E(X)2 where X is bet amount

E(X2) = 1002((0.0001)(4999)2+(0.9999)(-1)2) = 2500*104

V(X) = 2500*104 - (-50)2 and S(X) = 4999.75

standard error= 4999.75/10 = 499.975

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