Problem 3. A slot machine has 3 wheels, each with 20 symbols equally spaced arou

Question

Problem 3. A slot machine has 3 wheels, each with 20 symbols equally spaced around the wheel. On each play, the 3 wheels spin independently and each wheel is equally likely to show any one of its 20 symbols when it stops spinning. On the central wheel, 9 of the 20 symbols are bells, whereas the left and right wheels only have 1 bell each. The machine pays out the jackpot only if all wheels come to rest showing a bell. (a) What is the probability of hitting the jackpot? (b) What is the probability of hitting two bells but not the jackpot? (c) Suppose that instead there were 3 bells on the left, 1 in the middle, and 3 on the right. How would this affect your answers to (a) and (b)? Explain why the casino might find the 1-9-1 machine more profitable than the 3-1-3 machine.

Explanation / Answer

a) probability fo hitting a jackpot =P( 1st , 2nd and 3rd showing bells) =(1/20)*(9/20)*(1/20)=9/8000

b) probability of hitting 2 bells =P(  1st , 2nd showing bells and 3rd not showing+1st , 3rd showing bells and 2nd not showing+ 2nd, 3rd showing bells and 1st not showing)

=(1/20)*(9/20)*(19/20)+(1/20)*(11/20)*(1/20)+(19/20)*(9/20)*(1/20) =353/8000

c)for 3-1-3 machine ; probability of hitting jackpot =P( 1st , 2nd and 3rd showing bells) =(3/20)*(1/20)*(3/20)=9/8000

probability of hitting 2 bells =P(  1st , 2nd showing bells and 3rd not showing+1st , 3rd showing bells and 2nd not showing+ 2nd, 3rd showing bells and 1st not showing)

=(3/20)*(1/20)*(17/20)+(3/20)*(19/20)*(3/20)+(17/20)*(1/20)*(3/20) =273/8000

for 1-9-1 machine is showwing 2 bells more often ; therefore the customer is willing to bet more and more despite having the probability of getting a Jackpot is same in both the configuration

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