Analysis · 2026-10-04
California SuperLotto Plus: A Guide to Number Distributions
Understand the 5/47 + 1/27 format, theoretical frequencies, parity and sorted positions, with reproducible calculations and a historical-data checklist.
Correction, October 4, 2026: removed unsourced tables described as observations from 1,000 and 500 draws, corrected the DS Reports link, and replaced empirical-sounding claims with explicit theoretical calculations. This article does not present an analyzed historical dataset.
What a Distribution Study Can Establish
California SuperLotto Plus offers a useful example of the difference between a probability model and an observation. A model describes what fair draws would produce across repeated experiments. An empirical study measures a specified set of actual results. One cannot substitute a plausible-looking frequency table for the other.
This guide develops the model and a practical method for auditing future historical work. All numeric tables below are theoretical, not observed results. No date range, draw count or data-source citation is being claimed for a completed historical study. Records explored through DS Reports should be reconciled with the lottery operator before being described as verified evidence.
Rules and the Correct Sample Space
The California Lottery’s official SuperLotto Plus page specifies five main numbers from 1 to 47 and one separate Mega number from 1 to 27. The main numbers are distinct and their order is irrelevant to matching the main combination. The same printed number may occur in both pools because the pools are separate.
Write C(n, k) for the number of unordered k-element subsets of n distinct values. Under uniform main-number selection and an independent uniform Mega number:
- Main-number combinations: C(47, 5) = 1,533,939
- Complete main-plus-Mega outcomes: C(47, 5) × 27 = 41,416,353
- Jackpot probability of one specified complete line: 1 / 41,416,353
Matching the five main numbers alone is a different event from matching five plus Mega. Historical rankings of numbers do not alter either probability in a fair, independent next draw.
Individual Frequencies and Their Denominators
Across D complete draws, the 47 main-number counts sum to 5D and the 27 Mega counts sum to D. A fixed main number appears with probability 5/47 per draw. A fixed Mega number appears with probability 1/27. Dividing a main count by D measures the fraction of draws containing that number; dividing it by 5D measures its share of all main-ball entries.
Assuming independent draws and unchanged rules, the count of one fixed number across D draws is binomial. Its mean is Dp and its standard deviation is sqrt(Dp(1 − p)); see the NIST binomial reference. The following uses a hypothetical D = 1,000, solely to calculate expectations:
| Quantity | Per-draw probability | Expected count | Standard deviation |
|---|---|---|---|
| One specified main number | 5/47 | 106.38 | 9.75 |
| One specified Mega number | 1/27 | 37.04 | 5.97 |
These means are not quotas or predictions of the next result. Counts of different main numbers within one draw are dependent because sampling is without replacement. Comparing many numbers or many overlapping time windows also creates opportunities for apparently unusual results; a selected extreme is not automatically evidence of bias.
Number Ranges Are Unequal Groups
A range containing g of the 47 main numbers has expected share g/47 of all main-ball entries and expected count 5g/47 within one draw. Neither quantity is the probability of at least one member of the range appearing. That probability is 1 − C(47 − g, 5) / C(47, 5).
| Main-number range | Group size | Expected share of main-ball entries |
|---|---|---|
| 1–10 | 10 | 21.28% |
| 11–20 | 10 | 21.28% |
| 21–30 | 10 | 21.28% |
| 31–40 | 10 | 21.28% |
| 41–47 | 7 | 14.89% |
Rounding can make displayed percentages sum to slightly more or less than 100%. The last group is smaller, so equal group totals are not the correct baseline. For Mega groups 1–9, 10–18 and 19–27, each group contains nine numbers and has per-draw probability 9/27 = 1/3. Those are model expectations, not measured tier frequencies.
Parity and High/Low Categories
There are 24 odd and 23 even main numbers. Exactly r odd numbers in a five-number draw has probability C(24, r) × C(23, 5 − r) / C(47, 5). Counting the ways to select each part gives the following theoretical distribution:
| Odd numbers | Even numbers | Probability |
|---|---|---|
| 0 | 5 | 2.19% |
| 1 | 4 | 13.85% |
| 2 | 3 | 31.87% |
| 3 | 2 | 33.38% |
| 4 | 1 | 15.93% |
| 5 | 0 | 2.77% |
The middle categories contain more combinations. This does not make an individual balanced combination more likely than an individual all-even combination. Each specific main combination still has probability 1 / 1,533,939. Filtering tickets by parity removes possible outcomes rather than improving probability per ticket.
If “low” means 1–23 and “high” means 24–47, the two groups have 23 and 24 members. Exactly h high numbers therefore follows the same table as r odd numbers. State the threshold explicitly; changing it changes the category probabilities. No historical analysis is necessary to derive these values.
Sorted Positions Are Not Draw Order
A results page often lists main numbers in ascending order. Its first column is then the minimum, not the first ball extracted. For example, the minimum cannot exceed 43 in a five-from-47 draw because four larger distinct numbers must remain available. A histogram of that column is not uniform over 1–47 even when the draw is fair.
Retain original extraction order only if the source actually provides it. Do not infer extraction tendencies from sorted displays. Sorting is useful for duplicate detection and set comparison, but it changes the meaning of position-based statistics.
Reproduce the Calculations
This self-contained Python 3.8+ example uses only the standard library. It has no random sampling, downloaded data or hidden history. The output uses technical labels: main and mega identify the pools; odd identifies the number of odd main values. Python documents the combination calculation in math.comb.
from math import comb, sqrt
main_combinations = comb(47, 5)
jackpot_outcomes = main_combinations * 27
print("main_combinations:", main_combinations)
print("jackpot_outcomes:", jackpot_outcomes)
for name, probability in (("main", 5 / 47), ("mega", 1 / 27)):
mean = 1000 * probability
sd = sqrt(1000 * probability * (1 - probability))
print(f"{name}: mean={mean:.2f}, sd={sd:.2f}")
odd_counts = [comb(24, r) * comb(23, 5 - r) for r in range(6)]
assert sum(odd_counts) == main_combinations
for r, count in enumerate(odd_counts):
print(f"odd={r}: {100 * count / main_combinations:.2f}%")
main_combinations: 1533939
jackpot_outcomes: 41416353
main: mean=106.38, sd=9.75
mega: mean=37.04, sd=5.97
odd=0: 2.19%
odd=1: 13.85%
odd=2: 31.87%
odd=3: 33.38%
odd=4: 15.93%
odd=5: 2.77%
Checklist for an Actual Historical Study
- Save the official source URL, retrieval date, first and last draw dates, draw identifiers and rules period.
- Check five unique integers in 1–47 plus one Mega integer in 1–27 for every record. Keep the pools separate.
- Reconcile missing or duplicate draw records before calculating rates. Preserve zero counts for valid numbers that did not appear.
- Label every table as observed, theoretical or synthetic, and show the denominator beside each statistic.
- Compare observations with a preselected model. Account for dependence and multiple comparisons before making a statistical-significance claim.
- Publish the cleaned data or enough provenance to reproduce the calculation. Investigate discrepancies with official records before suggesting a game irregularity.
Theoretical agreement does not certify an unexamined archive, and a large historical sample does not create a selection advantage. Keep any lottery spending within a fixed entertainment budget; distribution analysis is not a promise of profit.
Educational content only. Past results do not predict future independent draws. DS Reports does not sell lottery tickets or guarantee outcomes.
Frequently Asked Questions
Does this article report a verified historical study?
No. The tables and code show theoretical calculations under a fair-draw model. No measured frequencies from a verified historical dataset are reported.
What is the SuperLotto Plus format?
Choose five distinct main numbers from 1 to 47 and one Mega number from a separate pool of 1 to 27. A complete line has jackpot probability 1 in 41,416,353 under the fair-draw model.
Why are main numbers and Mega numbers counted separately?
A fixed main number has inclusion probability 5/47 per draw, while a fixed Mega number has probability 1/27. Combining the pools would mix different events and denominators.
Are balanced combinations more likely than other specific combinations?
No. Parity categories contain different numbers of combinations, but every specific valid five-number combination has the same probability in a fair draw.
Where should historical results be verified?
Use the California Lottery’s official SuperLotto Plus results and rules. Record the date range, draw identifiers, missing records and rules period before calculating frequencies.