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# Even Odds?

## Question

I spent the afternoon in the garden, picking and shelling peas, collecting them in a large bag. When I got home my little sister reached into the bag and pulled out a handful of peas.

What is the probability that she pulled out an odd number of peas?

1. less than 1/2
2. 1/2
3. greater than 1/2

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Spoiler

Statistically speaking, it ought to be 1/2 ...unless there is a catch!

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Spoiler

It should be less that 1/2.

The difference lies in the plural use of “peas”.  To keep it simple let’s assume a moderate number of peas, say less than 7or 8   If 7 or less the amount could be  2,3,4,5,6,7    If 8 or less the amount could be : 2,3,4,5,6,7,8  that yields 7 even numbers and only 6 odd numbers  regardless of the upper limits, the odds always favor an even number

Edited by bonanova
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Without assuming a minimum number of peas in her hand, slightly greater than 1/2. There are two possibilities. You have an even number of peas or an odd number of peas (in the bag).

If you have an even number, the probability that she pulled out an odd number is 1/2.

If you have an odd number, there is one more odd number than even. So the probability is greater than 1/2.

Average the two and it's greater than 1/2.

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Do we get any information about how your sister pulls out the peas -- like should we act as if the number of peas that gets pulled out is a number drawn from a mean (perfect number of peas to fit in her hand) +/- standard deviation, a Poisson distribution, or something else -- or is that left for us to decide?

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14 hours ago, plasmid said:

Do we get any information about how your sister pulls out the peas -- like should we act as if the number of peas that gets pulled out is a number drawn from a mean (perfect number of peas to fit in her hand) +/- standard deviation, a Poisson distribution, or something else -- or is that left for us to decide?

Assumptions do matter. So here's some guidance, or not.

I tried to word the puzzle in a way that it could be reconstructed. Theoretically at least, a normal person could spend an afternoon shelling peas, with a normal person's productivity, put her normal work product into a bag, and then those peas could theoretically be counted, and a distribution made, or compared with various candidate distributions. Basically (and mainly because I'm not Bushindo) I don't know what it would be. Then, my fictitious little sister can be assumed to have a hand that has a holding capacity of one to two orders of magnitude smaller than the capacity of the bag. But whether she grabbed as many as possible, or not, is a random outcome. (Handful may not mean full hand. Some may have been a better word choice.) I have no idea what implications devolve from that. But one could do the experiment 1000 times, in principle, and look at the distributions. Or, the matter might be known. Just not by me. Then, jhawk's observation is valid. I hadn't intended it, but it's there in the OP so it should be taken into account.

If it helps, I have no problem with saying the total capacity of the bag is 10,000 peas; it was somewhere between 1/2 and 2/3 full; my sister's hand can hold no more than about 1% of the peas in the bag, and she may or may not have taken her maximum capacity of peas. Who understands sisters, anyway?

Finally, if there are plural considerations that lead to different answers, I believe one of them predominates.

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Spoiler

She will be either more likely to pull out an odd number of peas or an even number of peas, but we don't currently have enough information to know which probability is greater.

I'm assuming that there's a certain number of peas that are "just right" to fit in her hand (not necessarily an integer) and she'll pull out a number of peas with a probability in more or less a normal distribution around that "just right" number. In reality it's not a completely normal distribution because she can't pull out less than zero peas or greater than 10,000 peas, but going with a normal distribution is probably fine for the purposes of this question.

In an extreme case, if the standard deviation of that normal distribution approaches zero, she'll almost always pull out the number of peas that's comfy for her hand and the answer will be either even or odd depending on whether that number is even or odd. (Unless the number is exactly an integer plus 0.5 in which case she'll always pull out even and odd with equal probability because of symmetry, but the probability of a random real number being exactly an integer plus 0.5 is zero so I'll just ignore that case.) As the standard deviation grows, the probability of pulling even or odd numbers of peas quickly approaches 50/50, but probably never quite completely settles on it.

I did a little playing around with an OpenOffice worksheet using the norminv function to see if there are any values for standard deviation that would make the probability flip from favoring even to odd - conceptually, imagine if the bell curve widened just enough so the combined probability for the two numbers flanking the mean ever exceeded the probability of the mean itself - but I couldn't detect such an event.

It might also be worth saying how big of an effect that would be.

Spoiler

When I run 10,000 trials of pulling handfulls of peas with a mean pea number of 50 and various standard deviations, she pretty much always gets an even number of peas if the standard deviation is 0.1 or lower. With a standard deviation of 1, the even/odd effect is pretty small and I'm typically seeing even numbers of peas pulled about 51% of the time. Much more than that, and it gets tough to see a difference between 50/50 odds.

For reference of how much a standard deviation of 1 means, here's a histogram of the number of peas pulled in 10,000 trials with mean 50 and standard deviation 1.

Edited by plasmid

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