Question

A slot machine have an enterprise of 5$, it has four slots that generates numbers between number 1 and 5. If all the slots generates 5, you win 100$. If three of the slots generates 4, you win 20$. If two of the slots generates 3, or 2, you win 2$. Otherwise, you win x$. What is the smallest number that boss can set in order to profit on average?

Answer 1

Given that there are total 4 slots and each slot can generate numbers between 1 and 5

So there will be total of 5*5*5*5 = 625 outcomes

Case 1 : If all the slots generates 5, you win $100

This is possible only in one outcome where all the slots generate 5. So the probability of it is 1/625 = 0.0016

Net Profit in this case for the boss = 5 -100 = $-95

Case 2 : If three of the slots generates 4, you win 20$

This is possible when the three slots generate 4 and the otther slot can generate 1,2,3 or 5. So the other slot can take 4 values.

So the total outcomes possible where three of the slots generate 4 is 4/625 = 0.0064

Net Porfit  in this case for the boss = 5 - 20 = $-15

Case 3 : If two of the slots generates 3 or 2, you win $2

If two of the slots generates 3, then the other two slots can generate any of the four number other than 3. So there will total 4*4 outcomes. So probability of it is 16/625

If two of the slots generates 2, then the other two slots can generate any of the four number other than 2. So there will total 4*4 outcomes. So probability of it is 16/625

So probability that the two of the slots generates 2 or 3 is 32/625 = 0.0512

Net Profit in this case for the boss = 5 -2  = $3

Case 3 : Otherwise you win $x

Probability that Cases 1,2 or 3 doesn't happen = 1 - (0.0016 + 0.0064 + 0.0512)

= 1 - 0.0592

= 0.9408

Net profit in this case fort he boss = 5 - x

Expected profit for the boss = Probability in each case * profit in each case

= 0.0016 * (-95) + 0.0064 * (-15) + 0.0512 * (+3) + 0.9408 * (5 - x)

For the boss to be in profit, his expected profit should be greater than 0

So 0.0016 * (-95) + 0.0064 * (-15) + 0.0512 * (+3) + 0.9408 * (5 - x) > 0

- 0.152 - 0.096 + 0.1536 + 4.704 - 0.9408*x > 0

-0.9408 * x + 4.6096> 0

4.6096 > x * 0.9408

x < 4.6096 / 0.9408

x < 4.88966

The smallest number that the boss can set here is somewhat misleading, becuase it will be the lagest number that he can set which is 4.88966

If it is for the player to be in profit and not the boss, then the smallest value of x is 4.88966

Please let me know in comments if you have any doubts, i am ready to resolve them