Calculating lottery probability

[Mugged]

Hello, this lottery problem has been bothering me for months. I set it up here. Help appreciated.

Here is the setup:

Suppose you have to pick 5 numbers from a set of integers ranging from 1 to 39. The probability of matching x/5 (where x can be 1,2,3,4,5) numbers correctly can be found by computing the hypergeometric distribution formula, as found on wikipedia. we have that:

probability of matching x/5 = {5 choose x}*{(39-5) choose (5-x)} / {39 choose 5}

where {a choose b} = a! / (b!*(a-b)!)

Computing for x = 2,3,4,5 yields:

x=2: probability = 59840/575757 ≈ 0.104
x=3: probability = 1870/191919 ≈ 0.00974
x=4: probability = 170/575757 ≈ 0.000295
x=5:probability = 1/575757 ≈ 1.737e-6

Now here is the problem:

Suppose you buy 8 tickets, assume all of them have distinct sets of 5 integers, and play these 8 combinations for 1,000,000 drawings of the lottery. How many tickets in total (out of 8*1,000,000) do you expect to match x out of 5? i.e. in 8,000,000 plays, how many tickets yield 3 out of 5 matched?

I had originally thought that the solution was simply the probability of matching x/5 multiplied by the total number of plays, in this case 8e6, but this isn't the case. Although, this approach works for 2 out of 5 matches but not for 3 or 4...unsure about 5/5.

Any help is well appreciated.

 

[mfb]

If you know the expected hits (like the probability of "3 out of 5") for each ticket, the expected number of total hits is just the sum of all those expectation values - in your case, 8 million times the probability that a single ticket wins.

The variance is a bit more tricky, if those 8 tickets are not independent, but I think this can be neglected for every reasonable calculation.

 

[Mugged]

Right, but that approach doesn't seem to work. The reason I'm posing this question is because I've been running lottery simulations in Matlab and I'm getting results that are inconsistent with that idea.

I run 1,000,000 "games", where for each game I have 8 sets of 5 numbers chosen being compared to a winning game combination. Out of the 8 million plays, I get roughly 800,000 2/5 matches, 40,000 3/5 matches, can't remember 4/5,5/5.

But the point is the probability for 2/5 match is rougly 10%, so 8e6*.10 = 800k, which is fine. But for 3/5 match the probability is 0.009, times 8e6 is about 80k, but I only get 40k...so this doesn't match

 

[mfb]

I would expect a bug in the code then.

 

[blue_raver22]

Hello,

Thank you for providing the setup and your thoughts on this lottery probability problem. It seems like you have a good understanding of the formula and how to calculate the probability of matching a certain number of numbers correctly. However, your approach for calculating the expected number of tickets that will match a certain number of numbers out of 5 is not correct.

The correct approach is to use the binomial distribution formula, which takes into account the number of trials (in this case, 1,000,000) and the probability of success (the probability of matching x out of 5 numbers). So for example, to calculate the expected number of tickets that will match 3 out of 5 numbers, you would use the formula:

Expected number of tickets = (number of trials) * (probability of success)^3 * (1 - probability of success)^2 * (number of combinations)

In this case, the number of combinations would be 8 choose 3, since you are choosing 3 tickets out of 8. So the final formula would be:

Expected number of tickets = 1,000,000 * (0.00974)^3 * (1 - 0.00974)^2 * {8 choose 3}

I won't do the calculation for you, but this is the correct approach for calculating the expected number of tickets that will match a certain number of numbers out of 5. This approach can be used for any number of matches, not just 3 out of 5.

I hope this helps and good luck with your lottery calculations!