On Halloween night, a young boy was trick-or-treating. It was late at night and he was on his last stop. He knocked at the door and said:


The man inside said he had a bag of just 4 candy left. He said there were 1 hard candy, 1 gummy, and 2 lollipops.

He allows him to take 2 candy, and the boy does so. He grabs 2 from the bag without looking! He then states he has at least 1 lollipop, afterwards he asked his mom what the chances are of him having 2 lollipops.

His mom had no idea, but I bet you do! What were the chances of the boy pulling out 2 lollipops that night?

  • 3
    $\begingroup$ Doesn't this break this idea for what makes a good puzzle? Seems like a simple calculation of probabilities. $\endgroup$ – Rand al'Thor Dec 21 '14 at 15:25
  • $\begingroup$ @rand al'thor There's a reason it's a puzzle! Don't ruin it! $\endgroup$ – warspyking Dec 21 '14 at 15:28
  • $\begingroup$ @rand Answer it if it's so easy $\endgroup$ – warspyking Dec 21 '14 at 15:29
  • 2
    $\begingroup$ Don't mind if I do! :-) $\endgroup$ – Rand al'Thor Dec 21 '14 at 15:32

The answer is

1 in 5.

because given that he has at least 1 lollipop, the possibilities are:

  • 1 lollipop, 1 hard candy (probability 1 in 8)

  • 1 lollipop, 1 gummy (probability 1 in 8)

  • 2 lollipops (probability 1 in 16)

So the conditional probability of him having 2 lollipops is $\frac{1/16}{\frac{1}{8}+\frac{1}{8}+\frac{1}{16}}=\frac{1}{5}.$

At the OP's behest, I'm also explaining why it looks like the answer should be 1 in 3. (If anyone's not confused already, let's confuse 'em! :-) ) The boy has 1 lollipop; there are 3 possibilities for the other sweet, of which only 1 is 'the other lollipop'; so it looks like the answer should be 1 in 3. But we know nothing to distinguish the two lollipops. He has 1, yes, but which one did we choose? If the lollipops were one red and one yellow, and we knew he had the red one, then the answer would be 1 in 3. As it stands, however, the answer is 1 in 5 as proved above.

  • $\begingroup$ Snap... I was hoping I'd get the answer "1/3" before the real one lol! $\endgroup$ – warspyking Dec 21 '14 at 15:40
  • $\begingroup$ Would you mind explaining why it is not 1/3? $\endgroup$ – warspyking Dec 21 '14 at 15:41
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    $\begingroup$ @warspyking What do you mean? If it is 1/5, it can't be 1/3! $\endgroup$ – Rand al'Thor Dec 21 '14 at 15:43
  • $\begingroup$ Explain why some would think it is 1/3, and why that's incorrect... $\endgroup$ – warspyking Dec 21 '14 at 15:44
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    $\begingroup$ If I may chime in here, @warspyking, many of your posts read to me as if you thought of them immediately before posting -- they don't seem thoroughly analyzed. While interesting on their face, they could almost certainly benefit from your taking some time to think about how puzzlers are going to solve them, what edge cases they're going to find, what difficulties they're going to encounter, etc. I'd suggest having at least one other person read a post -- or just sit on it yourself for a day or so -- before issuing it as a challenge. Or post your idea for critique, not solving, here. Good luck! $\endgroup$ – jscs Dec 21 '14 at 20:11

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