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?