## Lotto, detailed.

The lotto page has been updated. The focus has been maintained; the point was to examine the common Lotto 6-49.

Taking for example a hypothetical Lotto 4-15, we select 4 numbers of our choosing from the available numbers ranging between 1 and 15. The 4 you pick are matched against another 4 random numbers drawn from the same range of numbers, 1 through 15, at a later date. You all know how a lottery works.

The point being, you want your 4 numbers to match the 4 numbers later drawn at random. But what are your chances of winning with 4 numbers picked from a pool of 15? Google says 15 choose 4 equals 1365. What’s that 15 choose 4 thing? It’s a binomial coefficient (n choose k), but I won’t get into too much detail about that here. Suffice it to say, it calculates the number of combinations to be made when choosing *k* numbers from a pool of size *n*.

So with 4 numbers from a total 15 you have a 1 in 1365 chance of having your 4 numbers picked again.

We can simulate that. The source code is right here. It’s the lotto sim. I set it to pick 4 numbers of it’s choosing from 15 and hold on to them. I also told it to simulate and record stats for 1000 wins.

So the sim chooses the 4 numbers it’s going to play then it picks 4 more, like a lotto-drawing would. If it matches, it’s happy and stops playing. If it doesn’t match then it plays the 4 numbers it originally chose against a new drawing of 4 (think of it like next week’s drawing). It keeps going until it finds a match and when it does it stops and records the number of games played until it won. Then it picks another 4 and plays until it wins… (based on number of wins you told it to play to).

Here’s a spiffy graph showing the Lotto 4-15 with statistics for 1000 wins.

1000 wins gives us an “ok” real-world result for the lotto 4-15. We see that the average starts to converge on the odds of winning this type of lottery. Remember 15 choose 4 gave us the value 1365 and the results here are showing a value of 1380.69 which is reasonable.

Click the **Science** tab at the top of this page and then the **Lotto** section. Visit the science page in the wiki. There you can find more information along with results for the common Lotto 6-49.