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The Minecraft wiki says (here) that, on average, under ideal conditions, after 31 minutes and 3.14 seconds, 80% of the crops in a wheat field should be fully grown.

I tried to prove that, but my result was 34 minutes and 55.55 seconds. I will list my calculations and ask for your help to find mistakes.

Per tick (there are 20 ticks per second), 3 individual blocks out of $16*16*16=4096$ are randomly chosen and "ticked", so per tick the probability for a single block to be random-ticked is $\frac{3}{4096}$. Each time a wheat crop is random-ticked, the chance for it to increase its age by 1 is $\frac{1}{3}$. Combined, the chance for a wheat crop to increase its age is $\frac{3}{4096}*\frac{1}{3}=\frac{1}{4096}$ per tick. A wheat crop has 8 stages, so it must increase its age 7 times to be fully grown.

Now, with Excel, I used the cumulated binomial-distribution to find the number of ticks where P(X>=k) is closest to 0.8. The Excel formula I used is =1-BINOM.VERT(7;n;1/4096;TRUE) (I tried multiple values for n).

n k p P(X>=k)
41910 7 1/4096 0.799982
41911 7 1/4096 0.800003
41912 7 1/4096 0.800023

The result closest to 0.8 is the one with 41911 ticks. In minutes, this is $\frac{41911}{20*60}=34.93$.

Did I make mistakes?

Silas
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  • When you write: " so the probability is 3/4096": probability of what? Then, you write "Combined, the chance for a wheat crop to increase its age is 3/4096 1/3 = 1/4096"*: What makes you think you can simply multiply here? Note that from what you describe, it's possible that up to 3 crops "grow" during a single tick. You don't account for this. – Jean-Claude Arbaut Nov 03 '21 at 21:10
  • @Jean-ClaudeArbaut I edited this part to make clear what I mean.

    If the multiplication is not allowed here, how would I do it correctly?

     – Silas Nov 03 '21 at 21:15

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