Hello everyone! I hope you are having a good day, anyways your day is going to get even better because you have the honor to learn about the Wheat Seed Problem today.
It is a mathematical riddle from stochastics (statistics) which asks the question “what is the likelyhood of a wheat population to ever go extinct?”
Let me introduce the basics: You are a farmer and start out with a single wheat seed. When you plant a seed into the ground, it grows into a wheat plant, which you can harvest. Harvest gives you anything from 0 to 3 (inclusive) new wheat seeds, which you can then plant again next year.
Let me show you an illustration:
(Icons from Mineclonia, https://codeberg.org/mineclonia/mineclonia)
Obviously, when you run out of seeds, you have nothing to replant the next year; The population goes extinct and the game is over.
The question is: What is the likelyhood of that never happening? In other words, what is the likelyhood that you can continue to plant seeds year after year, until infinity?
The probability for each plant (when harvested) are: 25% that it gives you 0, 1, 2, or 3 seeds each. That means that on average, you get 1.5 new seeds per plant, so you get more than you invested. This means that the average size of your population goes up with time.
However, whenever the population reaches 0 seeds in total at any one year, then the game is lost.
What is the chance that you can continue to plant new plants each year indefinitely?
solution
I will post the solution in a follow-up post. Think about it yourself first :)
The probability that you will eventually have zero seeds is 100%, given enough iterations.
Each plant has a 0.25 probability of giving zero seeds, so the population as a whole has a 0.25_n_ probability of giving zero seeds, where n is the number of plants. This may become very small, but never 0, so given infinite repetitions you will eventually get the zero seed outcome, regardless of how many plants you have.
While your thought process is interesting, that solution is incorrect ;)
I will be curious to see your approach as I cannot see any other answer that holds for an infinite run. Unless I’ve misunderstood your question?
It is called gambler’s ruin.
https://en.wikipedia.org/wiki/Gambler's_ruin
Essentially it will go extinct eventually no matter what. Infinity has that problem but if we bound the problem and we can give the odds of failure or the minimum number of seeds required to ensure success to a certain number of generations.
I am well aware of that. However, note that this is only true if the expectation value per round is zero, which is not the case here. As you get 1.5 seeds on average for each seed that you planted, the expectation value per plant here is 0.5 (above what you invested); so the theorem is not applicable.
Unfortunately, you cannot apply a statistical “expected earnings” calculation here, as the result must be discrete, i.e. you can only get an integer number of seeds from each plant, and thus from the entire crop. Instead, you need to look at the distribution of the total number of seeds returned after the harvest, from 0 to 3_n_. It’s going to resemble something like a binomial distribution, but I haven’t calculated it to get the exact result as all we actually care about is the binary: did we get 0 seeds. It is clear that, regardless of the value of n, there is a non-zero probability of getting 0 seeds, and thus losing. As I mentiobed in my other comment, you can calculate the probability of a particular round leading to loss as 0.25_n_. Whilst minute for large n, it’s still greater than zero, and so has a 100% chance of occuring when sampled an infinite number of times.
It is perhaps most obvious when we only have a few seeds. Condider the initial case, where we have 1 seed. At harvest we have a 25% chance of getting 0 seeds, and thus failing. If we have 2 seeds, both must give 0 for us to lose, with a probability of 25%25% or 0.250.25 giving a probability of 0.0625, or 6.25%. With three seeds, the probability of failure is 0.015625, or 1.5625%. Four seeds: 0.00390625, or approximately 0.39%. This demonstrates that a) you might not even get past the first round, and b) the probability of losing constantly drops with increasing n, but testing that an infinite number of times will, eventually, lead to the loss condition.
unlikely events occur routinely in infinite series. So the odds are all plants producing zero seeds is unlikely and even more unlikely to occur multiple times in a row but no matter how unlikely, the inevitable outcome will be it occurs and then you are down to zero.
