bowl of nuts

[Fusion_power]

This is not so much of a puzzle as it is a question about probability.

There is a large bowl of mixed nuts in front of you. The nuts are varying sizes and shapes including pecans, cashews, brasil nuts, etc. The smallest nuts in the bowl are peanuts and they comprise 50% of the total.

You reach into the bowl with your human size forefinger and thumb and pull out one nut. What is the probability it will be a peanut?


Before you make a fast answer, let me warn you that in actual practice, it is NOT 50%.

DarJones

[rogue]

Originally Posted by Fusion_power

This is not so much of a puzzle as it is a question about probability.

There is a large bowl of mixed nuts in front of you. The nuts are varying sizes and shapes including pecans, cashews, brasil nuts, etc. The smallest nuts in the bowl are peanuts and they comprise 50% of the total.

You reach into the bowl with your human size forefinger and thumb and pull out one nut. What is the probability it will be a peanut?


Before you make a fast answer, let me warn you that in actual practice, it is NOT 50%.

DarJones

What does mixed mean (beyond the obvious)? Does it mean that the nuts are somewhat evenly distributed or did all of the peanuts manage their way to the bottom of the bowl? What is the shape of the bowl? Where are you grabbing from, closer to an edge or closer to the middle? Are you grabbing from the top or from the bottom of the bowl or in the middle? Doesn't the answer also depend upon the kinds of nuts and the percentage of each? Have the nuts been shelled?

All I'm saying is that I highly doubt that the answer is fixed.

[Fusion_power]

whether it is near the middle or near the edge that you get a nut should not make a significant difference. Think about the way your thumb and forefinger work and you will see why.

The nuts are truly mixed meaning various sizes and species are spread throughout the bowl.

The objective is to pick a single nut out of the bowl without consciously 'selecting' by feel or by sight. Just let whichever one your fingers get be the one you remove.

Think about this a while and you will find that it is a very interesting problem. The size of your fingers is the biggest single factor.

DarJones

[cheesehead]

Originally Posted by Fusion_power

The nuts are truly mixed meaning various sizes and species are spread throughout the bowl.

... which is not the "natural" distribution resulting from gravitation. If the various-sized nuts were simply shaken-up in the (covered) bowl beforehand, the small ones will tend to wind up at the bottom, with the large ones at top. Same with pebbles in an aquarium bowl.

The objective is to pick a single nut out of the bowl without consciously 'selecting' by feel or by sight. Just let whichever one your fingers get be the one you remove.

Reminds me: there's a method to get your raffle tickets picked more often than random when equal-size tickets are simply drawn by hand. It's scarily effective and subtle (and possibly unethical -- so I won't provide more detail, and have fudged one or more here) -- the first time I used it, my ticket was picked as one of 3 out of about 50-60. After I described it to friends at a company Christmas party (which I won't do again!), in the after-dinner raffle three out of four couples at our table won door-prizes, the most of any table (enough to draw comments from others).

(Yes, I know the Law of Small Numbers, but if you knew the secret ...)

There's a way for the ticket-drawer to minimize the effect of this method (and I advise all of you to see that drawings are done this way if you have any influence):

Thoroughly, and I mean thoroughly mix the raffle tickets before drawing. Do so in a way that doesn't just shuffle tickets without changing their orientations, but actually flips them, repeatedly, in relation to one another, such that no group of tickets remains together from start to finish. When you see one of those drawings from a big rotating drum that seems to mix the tickets -- that may not be good enough. The drum should have interior paddles, as in a clothes dryer, to help flip and scatter the tickets in relation to each other during the rotations, and should be rotated many, many times (not very rapidly, either) between insertion of tickets and the drawing.

[retina]

For the nuts, I presume 50% is measured by the number of individual pieces? If so, then peanuts will comprise less then 50% by volume. I think the probability of drawing a peanut will be related to the overall volume ratios in the bowl.

For the raffle tickets, it is well known that folded vs non-folded tickets changes the odds of being drawn. They mix differently when the bowl of stirred and often drawers will draw from certain locations. Leaving out the exact details in line with cheesehead's post, but I don't see why, it is easy to work out and not really a secret.

[Fusion_power]

The peanuts are 50% by weight. It is a minor quibble given that most mixed nuts are sold by weight. I thought carefully about including this in the original question then decided to let folks make an educated guess. They could be mixed by weight, by volume, or by count. Presume the peanuts weigh exactly 1/2 the total weight of the bowl of nuts.

DarJones

[Uncwilly]

Originally Posted by retina

For the nuts, I presume 50% is measured by the number of individual pieces?

I was thinking that.

Originally Posted by Fusion

The peanuts are 50% by weight.

That is almost as bad, they pack tighter, thus Cheesehead's ticket issue become more important.
If it was by intial volume (10 L of peantus, 10L of others, mixed then poured), then we are talking a better challenge.

Also, was the bowl vibrated at all (even subtlely) after the random mixed nuts were placed in it? This would bias away from the goobers.
They ain't even really nuts!

[retina]

So also with 50% by weight we still get a volume difference. Some nuts will be more dense by than others. I would guess that peanuts are quite dense so would comprise less than 50% by volume and thus have a lower expectation of begin picked.

[Fusion_power]

The lleast dense nuts in the mix are pecans. The most dense are brasil nuts. The mixture is random and relatively even throughout the bowl.

The interesting thing I found is that there was an interaction between the size of my fingers and the size of the nut I picked up at random. My fingers tended to pick up larger nuts about 70% of the time. The shape of the nuts played a part since I tended to pick pecans much more often than expected given that they are a very small part of the mix. There must be a mathematical rule that would explain this.

DarJones

[Nelson]

Are you sure it doesn't have something to do with Dar Jones likes Pecans? I usually come up with a cashew which I like almost as much as a Brazil nut. So I suppose the shape has a slight effect the brazil nuts tend to slip away if my mother(RIP) hasn't gotten them first. I rarely come up with a peanut.

nelson

[takvorr]

I am working with non profit charitable organizations on running raffles. They are heavily regulated by state. Most require the use of two part raffle tickets of which one part is placed in a bin and the winner is drawn. I think that in order to have perfetc randomness the drum has to be turned enought imes so that every ticket (let's say there are 1000) has to be in a difefernt place in the drum from whe the drum was first turned. I doubt this owuld ever happen so I question if the traditional way of selecting a raffle ticket is truely random and fair. The only people really making mnoney are the raffle ticket printers. Is there a mor erandom and firer way to have a raffle drawing?