Combinatorics: calculating Oz Lotto odds for divisions

In summary, Oz Lotto is a lottery game where players select seven numbers from a total of 45. Nine balls are drawn, with seven being winning numbers and two being supplementary numbers. The odds of winning vary for each division, with Division 4 having odds of 1 in 29,602 and Division 7 having odds of 1 in 87. There are different methods for calculating the odds, but both the original method and an alternative method result in the same answers for Division 4.
  • #1
Darkmisc
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Homework Statement
In Oz Lotto, balls are numbered 1 to 45. Nine are selected, seven of which are winning numbers and two being supplementary numbers. Players select seven numbers.

The odds of winning can be found here: https://www.lottoland.com.au/magazine/oz-lotto-everything-there-is-to-know.html

I tried calculating the odds, and get all of them right except for div 4 and 7. Could somebody please explain what I've done wrong?
Relevant Equations
[SUP]7[/SUP]C[SUB]5[/SUB] x [SUP]2[/SUP]C[SUB]1[/SUB] x [SUP]36[/SUP]C[SUB]1[/SUB]/ [SUP]45[/SUP]C[SUB]7[/SUB] =

1/30,012
In Oz Lotto, balls are numbered 1 to 45. Nine are selected, seven of which are winning numbers and two being supplementary numbers. Players select seven numbers.

The odds of winning can be found here: https://www.lottoland.com.au/magazine/oz-lotto-everything-there-is-to-know.html

I tried calculating the odds, and get all of them right except for Div 4 and 7. Could somebody please explain what I've done wrong?

Division 45 Winning Numbers + 1 Supplementary Number1 : 29,602

7C5 x 2C1 x 36C1/ 45C7 =

1/30,012
Division 73 Winning Numbers + 1 Supplementary1 : 87

7C3 x 2C1 x 36C3/ 45C7 = 1/91

Thanks
 
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  • #2
I think I agree with your answers!
 
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PS alternative calculation for division 4:
$$p = 7 \cdot 6 \cdot P(WWWWWSX) = 42 \cdot \frac{7}{45} \cdot \frac{6}{44} \dots \frac{2}{40} \cdot \frac{36}{39}$$
 
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Related to Combinatorics: calculating Oz Lotto odds for divisions

1. What is combinatorics?

Combinatorics is a branch of mathematics that deals with counting and arranging objects or events in a systematic way.

2. How do you calculate odds for Oz Lotto divisions?

To calculate the odds for Oz Lotto divisions, you need to use the formula for combinations: nCr = n! / (r!(n-r)!), where n is the total number of balls in the draw and r is the number of balls that need to be matched to win a specific division. For example, to calculate the odds of winning Division 1 in Oz Lotto, you would use the formula 45C7 = 45! / (7!(45-7)!) = 45,379,620.

3. What do the different divisions in Oz Lotto mean?

The different divisions in Oz Lotto refer to the different prizes that can be won based on the number of matching numbers. Division 1 is the highest prize and requires all 7 numbers to be matched, while Division 2 requires 6 numbers plus 1 supplementary number. The lower divisions require fewer matching numbers and offer smaller prizes.

4. How are the odds of winning different divisions in Oz Lotto determined?

The odds of winning different divisions in Oz Lotto are determined by the number of possible combinations of numbers that can be drawn and the number of combinations that will result in a win for each division. The more numbers that need to be matched, the lower the odds of winning that division.

5. Can you improve your chances of winning by using a specific strategy?

No, the odds of winning in Oz Lotto are purely based on mathematical probability and cannot be influenced by any strategy or method. Every number combination has an equal chance of being drawn, so the best way to increase your chances is to buy more tickets.

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