# Oz Lotto Odds -- Australian Lottery

TR Smith is a product designer and former teacher who uses math in her work every day.

Oz Lotto is one of the big national lotteries available in all states and territories of Australia. It is administered by Tattersall's lottery company and each game costs \$1.10 plus the agent's commission. During the Tuesday night drawings for Oz Lotto, seven numbers are drawn from between 1 and 45 inclusive, plus two more supplementary numbers from the same pool, for a total of nine numbers. The seven main numbers determine the prizes for Divisions 1, 3, 5, and 6; the supplementary numbers are counted in the prizes for Divisions 2, 4, and 7. Here is how to play and how to calculate the exact probability of winning each prize division.

## How to Play Oz Lotto

For each game you purchase you select 7 distinct numbers from 1 to 45. Even though lottery officials draw 9 numbered balls on Tuesday nights, you don't make any selection for the 2 supplementary balls. As with any lottery, there is no statistical advantage in picking or avoiding certain numbers.

## How to Calculate the Probability of Winning 1st Division

The 1st Division prize in Oz Lotto is awarded to players whose seven picks match all seven of the main numbers in the drawing. The probability of this occurring is the reciprocal of the total number of different game selections you can make. That total is (45 C 7), "45 choose 7," the combination function. This works out to

[45*44*43*42*41*40*39] / [7*6*5*4*3*2*1]
= 45,379,620

Therefore, the probability of winning the top prize is 1/45379620 = 0.0000000220363, aka a very small number. You are more likely to get kicked in the donger by a kangaroo than win the Oz Lotto jackpot. If you do win the jackpot, your winnings are approximately 40% of the prize fund.

## Probability of Winning 2nd Division

Calculating the odds of winning the 2nd Division prize is a little trickier because this prize is won by players who match 6 out of 7 main numbers and 1 of the 2 supplementary numbers. The probability of this happening is the total number of ways to make a 6/7 + supp match divided by the total number of Oz Lotto combinations. This is

(7 C 6)*(2 C 1)/45379620
= 7 / 22689810
≈ 0.0000003085085
≈ odds of 1 in 3241401.4286

You are still more likely to get kicked in the donger by a kangaroo than win the Division 2 prize.

## Probability of Winning 3rd Division

You win a Division 3 prize if you match 6 of the 7 main numbers but not one of the supplementary numbers. The probability of this event is the number of ways of picking 6/7 and not one of the supplementary, all divided by the total number of different lottery combinations. This is

(7 C 6)*(36 C 1)/45379620
= 7 / 1260545
≈ 0.0000055315
≈ odds of 1 in 180077.8571

The odds of getting kicked in the donger by a kangaroo are still greater than your odds of winning the Division 3 prize in Oz Lotto.

## Probability of Winning 4th Division

The 4th Division Prize in Oz Lotto is for players who match 5 out of 7 plus either 1 or 2 of the supplementary numbers. The calculation for this probability is a little trickier since you have to sum two distinct probabilities -- that of matching 5/7 and 1 supp, and that of matching 5/7 and both supps. This is

[(7 C 5)*(2 C 1)*(36 C 1) + (7 C 5)*(2 C 2)]/45379620
= 511 / 15126540
≈ 0.0000337817
≈ odds of 1 in 29601.8395

Here's where things get interesting; the likelihood of winning the Division 4 prize is the same as your chances of getting kicked in the donger by a kangaroo.

## Probability of Winning 5th Division

To win the 5th Division prize, you must match 5 out of 7 and none of the supplementary numbers. This probability is

(7 C 5)*(36 C 2)/45379620
= 147/ 504218
≈ 0.000291541
≈ odds of 1 in 3430.0544

At this point, you are more likely to win this prize than get kicked in the donger by a kangaroo.

## Probability of Winning 6th Division

To win the Division 6 prize, the second-lowest prize tier, you must match exactly 4 out of 7 numbers. Whether or not any of the other 3 nonmatching number is a supplementary number is irrelevant for this prize level. The probability of this happening is

(7 C 4)*(38 C 3)/45379620
= 4921 / 756327
≈ 0.00650644
≈ odds of 1 in 153.6938

## Probability of Winning 7th Division

Finally, the lowest prize division is matching 3 out of 7 and either 1 or 2 supplementary numbers. Following the calculation for the 4th Division prize, this probability is

[(7 C 3)*(2 C 1)*(36 C 3) + (7 C 3)*(2 C 2)*(36 C 2)]/45379620
= 17395 / 1512654
≈ 0.0114997
≈ odds of 1 in 86.9591

## Summary Of Probabilities

Prize Division
What to Match
Approximate Odds
1
7/7
1 in 45379620
2
6/7 + 1 supp
1 in 3241401
3
6/7 + 0 supp
1 in 180078
4
5/7 + 1 or 2 supp
1 in 29602
5
5/7 + 0 supp
1 in 3430
6
4/7
1 in 154
7
3/7 + 1 or 2 supp
1 in 87
overall
at least 3/7
1 in 55

## Total Probability of Winning Any Prize

Adding up all the probabilities from the seven prize divisions gives you a total prize probability of 4623/252109 ≈ 0.0183373, or approximately odds of 1 in 54.5336. Compared to other national lotteries in Australia, Oz Lotto offers a better overall chance of winning some prize. Australian Powerball's overall odds are about 1 in 110, while Saturday Lotto's odds are about 1 in 85.

## How to Increase Your Odds of Winning

The only way to increase your overall odds is to buy more games per drawing. In Australia, playing in syndicates (buying lots of tickets as a group) is a popular way to spread the cost of playing Oz Lotto over many tickets. Playing the lottery this way increases the likelihood of holding a matching ticket, though if your group does win, the prize will be split among all the members of a group.

When nobody holds a winning game slip for the jackpot, the amount rolls over to the next drawing, which means jackpots can grow very large when there are no winners for many drawings in a row. The largest ever Oz Lotto jackpot was for roughly \$112 million, shared among four winning tickets. One of the ticket holders was a syndicate of 80 members, two ticket holders were families, and one was a single player. Since the syndicate's share of the jackpot was 1/4, each member received approximately \$112000000*(1/4)*(1/80) = \$350,000.

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