The Riddle of the Mushroom

I was eating breakfast with my son after his Ju-Jitsu class at a local diner and had ordered a nice omelet with onions, bacon, spinach, cheddar cheese, and mushrooms. Seeing me eat a mushroom prompted a comment. He's been reading a book on surviving in the woods and he mentioned that if you are lost you should never attempt to eat a mushroom.

Firstly, because one needs calories and not protein to survived your escape from the woods, and secondly, because it's just to hard to tell a good one from a bad one. I told him that was probably true. He then mentioned that in different locales the types of poisonous mushrooms varied and that even if you got good in one area it might be a deadly mistake in another. Another good reason to avoid them.

This isn't to be fooled around with either. Mushroom poisoning is one of the most horrible deaths known to man and certainly not something you want to deal with when trying to get back to civilization. Eating the wrong one can result in vomiting, diarrhea, cramps, hallucination, and liver failure. So even if you are 99% sure you can make the correct identification then why take the chance.

Well this got me to thinking. What are the true risks involved. I thought of a another issue involved in eating mushrooms.  Humans have bad skills at estimating mathematical risk. I came up with this puzzle which actually has two levels at which natural human instincts on risk are likely to fail. One is known by statisticians but I’ve never heard the other expressed in a math puzzle. After someone has solved these I will point to a video by a statistician explaining the original problem.

Given:

You are lost in the wood on an alien planet. There are precisely two kinds of mushrooms here which look almost identical. One is edible and the other kills within seconds of even the smallest taste. Both kinds, unlike earthly mushrooms, provide plenty of carbohydrates to power your muscles for the long trip home.

Back at the lab on your spaceship at a different local with both kinds of mushrooms you had tested your ability to discriminate between mushroom types and have found yourself to have an accuracy rate of 99%. This is no matter how long you look at the same mushroom. You did so by comparing against a 100% accurate test that involves killing lots of mice, but let’s not let PETA in on this.

The forest floor is covered in millions of these mushrooms of both types. You are trying to decide whether to eat a mushroom. You want to know the exact risk of killing yourself if you eat a mushroom, vs. your estimated risk of starvation.

What is your risk of dying in the following independent scenarios? In each case you examine all picked mushrooms in detail to categorize them as “good” or “bad”. You only examine each mushroom once since your accuracy does not increase with time or repetition. The risk is always the risk of dying. The words bad and good refer to your estimation and not actuality.

1) You pick one mushroom. It looks good. What is your risk if you eat it?

2) You pick 10 mushrooms and they all look good to you.

a) If you eat a good one is your risk worse, better, or the same as case 1)

b) What is your risk if you eat just one?

c) What is your risk if you eat all ten?

3) You pick 1000. One looks good.

a) If you eat a good one is your risk worse, better, or the same as case 1)

b) If you eat it what is your risk?

4) You pick 200. You decide half are “good”.

a) If you eat one of good ones what is your risk?

b) If you eat all 100 good ones what is your risk?

5) You pick 10000. You decide that 100 are “good”.

a) If you eat one good mushroom what is your risk.

b) You’ve examined more mushrooms than in case 2) to come up with 100 mushrooms to eat. Are you safer or less safe eating all 100 in this scenario than in 2)?

c) What is your exact risk In b)?

6) You pick 10000 mushrooms. You decide that only one is bad. What is your risk if you eat the bad one?

I don’t know if I can come up with the exact answers in each case but I do know that some obvious answers are wrong.   I haven't been in a statistics class in around thirty years so I have an excuse.  No need to answer them all so try the easier ones.

My oldest son is doing a fair job of attempting to answer the questions. My younger son is calling me a fool for wasting all the mice which I should have brought with me to either eat directly or to use as a 100% accurate test.

Share this

Missing information

What is your prior distribution on the proportion of good mushrooms in the forest?

It varies by locale

It varies by locale and I don't know the distribution at the locale I am lost in.

What varies from locale to

What varies from locale to locale is the actual ratio of good to bad mushrooms, but the prior distribution of that variable, across many locales is yours to have. Even if you have no knowledge of the place you have to form at least a rudimentary prior, like uniform distribution of the percentage of good mushrooms over [0,1]

Without a prior, you cannot compute *any* probability.

"What varies from locale to

"What varies from locale to locale is the actual ratio of good to bad mushrooms, but the prior distribution of that variable, across many locales is yours to have."

Perhaps if you took a survey of lots of locations. That's not information you have. It doesn't matter if it's your to have because you are lost and starving and can't get it.

"Even if you have no knowledge of the place you have to form at least a rudimentary prior, like uniform distribution of the percentage of good mushrooms over [0,1]"

Why do you think that. You landed somewhere did your tests at the one locale and figured out how good you were at identifying mushrooms. You certainly don't know the worldwide distribution of the mushrooms based on that.

"Even if you have no knowledge of the place you have to form at least a rudimentary prior, like uniform distribution of the percentage of good mushrooms over [0,1]"

Why? Even if thought you had determined a uniform distribution worldwide that doesn't mean you couldn't just happen to be in a place that is outside the norm.

"Without a prior, you cannot compute *any* probability."

Really? You can't answer even answer question 1)?

you can't even answer question 1

1) The probability that the mushroom will kill you is
P(BAD|LOOKSGOOD) = P(LOOKSGOOD|BAD).P(BAD)/P(LOOKSGOOD)

since P(LOOKSGOOD) = P(LOOKSGOOD|BAD).P(BAD) + P(LOOKSGOOD|~BAD).(1-P(BAD)) = 0.01 * P(BAP) + 0.99 * (1-P(BAD)) = 0.99 - 0.98.P(BAD)
then P(BAD|LOOKSGOOD) = 0.01 * P(BAD) / (0.99 - 0.98.P(BAD))

You need a prior on P(BAD)

You can try to evaluate P(BAD) by sampling from the mushrooms but then you need a prior distribution of P(BAD) to correct for your estimation... say for example you look at 100 mushrooms and find 10% are bad but believe on no planet it's possible to have as low a probability then it means you have been mistaken => you need a prior distribution of P(BAD)

 

Back to the mushrooms, why would we start eating mushrooms ? The learning process is really risky, and the reward looks meager, only a few calories, why take the risk? One answer is that at some point we were really really despaired. Another crazy answer is that the reward was much higher, maybe we started trying mushroom for psychoactive effects and eventually stumbled on some dull ones that tasted not too bad. Yet another possibility is that we relied on animals/prisonners to try mushrooms.

Yeah I think you are right

Arthur,
I was thinking you could estimate the distribution from the actual test. But I think that makes it so that you cannot actually get a precise answer. Plus I don't know how valid it is. Certainly if you pick a lot of mushrooms, have a test that is .99 percent accurate, then you are going to have a general idea of distribution in some of the questions.

So I guess it's a good idea gone bad. Some things we know the distribution of like cards in a deck. But in this case you just can't know so you can't come up with an exact probability.

Certainly the risk in question 1) is around .01 I didn't try the calculation and thought it was going to settle out at .1, or if I used the test itself to approximate the distribution then close to that, but I wasn't sure how to correct for the very small sample size. Surely if I test a million mushrooms in a locale and 99% test white then there is a large percentage of whites.

I'll have to think about it more. Actually the assumptions in 6) seem implausible. If the mushrooms were all white in a locale and I picked 10000, then I'd expect to have around 100 good ones identified as bad.

It just seemed to me that I could use the results to estimate the distribution with larger numbers of picked but perhaps that's just wrong.

It just seemed to me that I

It just seemed to me that I could use the results to estimate the
distribution with larger numbers of picked but perhaps that's just
wrong.

You definitively can get an estimation, and with 99% accuracy you'll get a pretty good one... the prior distribution does affect your estimation, but to affect it seriously would take a silly prior.

Any prior distribution with a support = [0,1] will give the same result if you can look at enough mushrooms.