What are your chances of winning?

On the 3rd of May, 1997, I went into a lotto outlet and they gave me a printout of the number of winners for each division. The results were as follows:

For Division One there were 3  winners
For Division Two there were 17 winners
For Division Three there were 622 winners
For Division Four there were 27272 Winners
For Division Five there were 35335 winners

This provides a crude estimate of how likely it is to win each division. Division One is of course the hardest to win and as we move towards Division Five it becomes progressively easier. However in order to put things into more of a mathematical setting and thus obtain more accurate results I have calculated the probablities of a each division win. I have not provided the details of my calculations on this page but you can read them if you wish by following my web link.

Summary of Probabilities

Probability of a Division One Win
	= 1/3 838 380 
	= 2.6053 x 10^-7
	= 0.000026 %

Probability of a Division Two Win
	= 6 /3 838 380 
	= 1.5632 x 10^-6
	= 0.000156 %

Probability of a Division Three Win
	= 204 /3 838 380 
	= 5.3147 x 10^-5
	= 0.00531 %

Probability of a Division Four Win   
	= 8415 /3 838 380 
	= 2.1923 x 10^-3
	= 0.219 %

Probability of a Division Five Win
	= 10560 /3 838 380 
	= 2.7512 x 10^-3
	= 0.275 %

Now a 0% probability represents no chance of winning (a guaranteed failure) and 100% would represent a guaranteed success. Accordingly 50% represents an even chance. You will have noticed that these results look fairly pessimistic. However you must remember that these results are for one selection only. You cannot even take one selection on its own as the minimum price of a lotto ticket is $2.00 and this is for four selections.

As the number of selections purchased goes up so do your chances of winning. This is shown more clearly in the following graph. The green line shows results obtained by my proposed systematic approach and the blue line represents buying lucky dip tickets. For those interested the lucky dip graph is produced using the binomial distribution b(n;p) where p is the probability of single division one win and n is the number of tickets as shown along the bottom of this graph.

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