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Lotto 6/45 Draw 748: Winning Numbers and Probability Explained

Winning numbers and bonus number

  1. 3
  2. 10
  3. 13
  4. 22
  5. 31
  6. 32
  7. 29

The six winning numbers on 2017-04-01 were 3, 10, 13, 22, 31, 32 and the bonus number was 29. The bonus number only decides second prize, so it is left out of the number-feature calculations below.

First and second prize and the amount after tax

9 players won first prize, KRW 1,928,246,542 each (about KRW 1.93 billion). Applying the current tax rules, the amount after tax is about KRW 1,324,925,184 per winner. Second prize went to 59 players, KRW 49,023,218 each, about KRW 38,238,111 after tax.

Prize results of draw 748
PrizeWinnersPer winnerAfter tax (reference)
1st9KRW 1,928,246,542KRW 1,324,925,184
2nd59KRW 49,023,218KRW 38,238,111
3rd1,937KRW 1,493,222—
4th99,448KRW 50,000—
5th1,652,756KRW 5,000—

The after-tax figures apply the current rules (income tax withheld at 20%, 30% on the part above KRW 300 million, local income tax of 10% of that, nothing withheld at KRW 2 million or less) to this draw's prize with no purchase cost deducted. The rules in force at the time may have differed. The method is the same as in the prize tax calculator.

Sales in this draw were KRW 72,750,277,000. The five prize levels together paid KRW 36,375,139,754, which is 50.0% of sales. The share varies by draw, for example when a jackpot with no winner is carried to a later draw.

Was the number of first-prize winners ordinary?

72,750,277 games were sold in this draw. If every game were chosen independently at random, the expected number of first-prize winners would be 8.93 (games sold ÷ 8,145,060). The actual number was 9.

Comparison with the Poisson model
ExpectedActualP(X = 9)P(X ≥ 9)
8.93913.2%53.5%

9 winners is an ordinary result, close to what the model expects. A count of 9 or more has a probability of 53.5%, about once every 1.9 rounds.

The Poisson distribution is an approximation. In practice many people choose similar numbers, such as birthdays, so the real count varies more than the model says. The model probability is a yardstick only and says nothing about which numbers come up. See How many Lotto 6/45 jackpot winners should each round have? for the full explanation.

Cumulative comparison

Rounds 88 to 748 (661 rounds): the distribution of first-prize winner counts (bars) against the Poisson expectation for the same rounds (line).

First-prize winner counts versus the Poisson expectationBars are actual round counts; the line is the Poisson expectation: 0=3, 1=15, 2=29, 3=55, 4=80, 5=62, 6=87, 7=86, 8=75, 9=69, 10=27, 11=28, 12=12, 13=10, 14=8, 15=6, 16=4, 17=1, 18=1, 19=1, 20=1, 21=0, 22=0, 23=0, 24=0, 25=0, 26=0, 27=0, 28=0, 29=0, 30+=102449739705101520253030+First-prize winners in a round

The variance of the deviations measured in model standard deviations is 1.44 (1 under a Poisson model). The real counts varied more than the model, in line with the clustering of purchases described above.

Features of the six numbers and their exact probabilities

The probabilities below come from counting all 8,145,060 ways to draw six distinct numbers exactly (no approximation). "Probability of this outcome" is the share of all combinations that have the same feature as this draw.

Features of the numbers in draw 748
FeatureThis drawProbability of this outcome
Odd and even3 odd, 3 even33.5%
Low and high4 low (1–22), 2 high (23–45)22.7%
Neighbouring numbers1 neighbouring pair52.9%
Sum of the six111sum ≤ 111: 19.3%
sum ≥ 111: 81.6%
All six at most 31no (includes 32 or higher)91.0%

Every individual combination has the same 1 in 8,145,060 chance of being drawn. The shares above only show which shapes of combination are more numerous.

Context: how often each number was drawn

Appearances of each number from 1 to 45 among the six winning numbers (bonus excluded) over draws 1 to 748 (748 draws). If the draws are fully random, each number is expected 99.7 times with a standard deviation of about 9.3. The shaded band is the expectation plus or minus two standard deviations.

Appearances per number versus the theoretical expectationThe shaded band is the expectation plus or minus two standard deviations: 1=110, 2=99, 3=99, 4=107, 5=105, 6=92, 7=103, 8=106, 9=75, 10=103, 11=103, 12=97, 13=106, 14=106, 15=102, 16=91, 17=108, 18=103, 19=100, 20=116, 21=94, 22=84, 23=93, 24=97, 25=100, 26=100, 27=117, 28=87, 29=92, 30=87, 31=99, 32=88, 33=104, 34=117, 35=94, 36=98, 37=111, 38=93, 39=99, 40=116, 41=91, 42=90, 43=102, 44=102, 45=1020316192122151015202530354045Number

Chi-square goodness-of-fit test: χ² = 35.8, 44 degrees of freedom, p-value = 0.806. The p-value is at least 0.05, so the differences between numbers are within random variation.

Why this result tells you nothing about the next draw

Each draw starts from scratch as an independent trial. The drawing machine does not remember earlier rounds, so the numbers, the odd-even split, the sum and the winner count of this draw have no influence on the next one, and this result tells you nothing about it. Every combination has the same 1 in 8,145,060 chance in every draw. The figures on this page explain a draw that has already happened.

Source and collection time

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