I can't speak to quads, but I just ran the trips in Missouri's Pick 3 draw.
Over the entire history of the Missouri Pick 3, there have been 18,638 draws.
The odds of a trip being drawn are 1 in 100, so the expected number is 18,638 / 100 = 186 (rounded to whole number)
By my count, there have been 174 trips drawn in the entire history of the Missouri Pick 3.
You are right in thinking that is low, but is it out of bounds?
I asked Google: In 18,638 pick 3 draws, the expectation for trips is 186.38. Is an actual draw result of 174 trips out of bounds statistically?
Google answered:
No, an actual draw result of 174 trips is not statistically out of bounds. It falls well within normal statistical variations, sitting less than 1 standard deviation away from the expected 186.38 trips.
To determine if the variance is statistically significant, we can model the draws using a binomial distribution (or a normal approximation):
1. Identify the Variables
- Total Draws (\(n\)): \(18,638\)
- Probability of a trip (\(p\)): \(0.01\) (Since there are 10 triples—000 through 999—out of 1,000 possible combinations in Pick 3)
- Expected Value (\(E\)): \(186.38\)
- Actual Result (\(X\)): \(174\)
2. Calculate the Variance and Standard Deviation
The standard deviation (\(\sigma \)) measures the typical variation you can expect from the average by chance:
\(\text{Variance\ }(\sigma ^{2})=n\times p\times (1-p)\)
\(\sigma ^{2}=18638\times 0.01\times 0.99=184.5162\)
\(\text{Standard\ Deviation\ }(\sigma )=\sqrt{184.5162}\approx 13.58\)
The Z-score reveals how many standard deviations the actual result (\(174\)) is from the expected mean (\(186.38\)):
\(Z=\frac{X-E}{\sigma }\)
\(Z=\frac{174-186.38}{13.58}=\frac{-12.38}{13.58}\approx -0.91\)
In statistics, a result is generally considered "out of bounds" (statistically significant) if the Z-score is less than \(-1.96\) or greater than \(+1.96\) (which corresponds to a 95% confidence level).
Because your Z-score is only \(-0.91\) (with a two-tailed p-value of approximately \(0.362\)), this deviation is very common. There is about a 36.2% chance of seeing a result this far or farther from the expectation purely due to random luck.
I don't have the numbers for all the other lotteries to compare, but the Missouri Pick 3 trips rate is not statistically out of bounds.