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BMAD

I want the butterscotch

Question

There is a machine with 20 pieces of candy.  Five of those candies are butterscotch.  If you put in a 25 cents, one candy is provided at random.  If you put in 75 cents, two candies are dropped at random but you may give the machine back one candy in exchange for a 25 cents.  And if you put in $1.50 you receive 5 pieces of candy at random but are guaranteed at least one butterscotch.

How much should I expect to spend to get all of the butterscotch?

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6 hours ago, BMAD said:

There is a machine with 20 pieces of candy.  Five of those candies are butterscotch.  If you put in a 25 cents, one candy is provided at random.  If you put in 75 cents, two candies are dropped at random but you may give the machine back one candy in exchange for a 25 cents.  And if you put in $1.50 you receive 5 pieces of candy at random but are guaranteed at least one butterscotch.

How much should I expect to spend to get all of the butterscotch?

Clarification question:

If I put in 75p, can I return a candy I received from a previous purchase for the 25p rebate?

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14 hours ago, Molly Mae said:

Clarification question:

 

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If I put in 75p, can I return a candy I received from a previous purchase for the 25p rebate?

 

Yes, the machine only requires that one of its candies be returned it need not be one purchased with the 75 cents.

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Two more clarification questions

Spoiler

First, I see no reason to ever use the 75 cent option. If you pay 75 cents to get two pieces at random and you don’t return a piece, then you would have been better off using the 25 cent option twice and spending 50 instead of 75 cents. If you pay 75 cents to get two pieces at random and do return a piece, then you would have only spent 50 cents, but you would have gotten two random pieces and returned a non-butterscotch piece. You would have been better off paying 25 cents twice to get two random pieces without returning a non-butterscotch, because if you do return a non-butterscotch then that just decreases your chances of getting a butterscotch on subsequent rounds. If that analysis is wrong because of an incorrect assumption then let me know.

Second, I think it could make a difference exactly how the 1.50 option works. Specifically, if the machine would have randomly pulled out five non-butterscotch candies, would it drop one of them and pick up a butterscotch? Or would it drop all of them back into the bin and mix all the candies still in the bin and then draw five more random candies? Or would it set the five that it picked aside and draw five more candies that were not in its initial draw before dumping everything back into the bin?

 

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18 minutes ago, plasmid said:

Two more clarification questions

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First, I see no reason to ever use the 75 cent option. If you pay 75 cents to get two pieces at random and you don’t return a piece, then you would have been better off using the 25 cent option twice and spending 50 instead of 75 cents. If you pay 75 cents to get two pieces at random and do return a piece, then you would have only spent 50 cents, but you would have gotten two random pieces and returned a non-butterscotch piece. You would have been better off paying 25 cents twice to get two random pieces without returning a non-butterscotch, because if you do return a non-butterscotch then that just decreases your chances of getting a butterscotch on subsequent rounds. If that analysis is wrong because of an incorrect assumption then let me know.

Second, I think it could make a difference exactly how the 1.50 option works. Specifically, if the machine would have randomly pulled out five non-butterscotch candies, would it drop one of them and pick up a butterscotch? Or would it drop all of them back into the bin and mix all the candies still in the bin and then draw five more random candies? Or would it set the five that it picked aside and draw five more candies that were not in its initial draw before dumping everything back into the bin?

 

To answer your second part: the machine would scan the candies and intentionally pick out a butterscotch and randomly select four candies from the remaining four. So in this case, if there is a butterscotch candy left then you are guaranteed that the first one chosen was a butterscotch. 

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