Can you solve this very tricky lottery related math problem?

[MegaMillions]

So, last night was a very odd drawing for the Mega Millions game.

The odds of matching the 5 white balls are 1 in 3,819,816:

There are 56 balls. 56 x 55 x 54 x 53 x 52 = 458,377,920. Since you don't have to match them in a specific order, divide that by the number of ways in which 5 numbers can be organized (5 x 4 x 3 x 2 x 1), which is 120.

That makes the odds of matching the 5 white balls 1 in 3,819,816.

In the 4/17/2009 drawing, for a random example, there were approximately 33,785,200 tickets sold. 10 tickets matched the 5 white balls. 33,785,200 divided by 3,819,816 = about 9.15. So 10 winning tickets is right about average. Normally the number of winners for each respective category closely follows the odds. In most drawings, about every 3.8 million tickets will be a ticket that matches the 5 white numbers.

If an unusual amount of tickets wins a given prize, this is how I calculate the odds of it happening:

In the 5/1/2009 drawing, there were approximately 78,130,400 tickets sold. The odds of winning the jackpot are 1 in 3,819,816 x 46, since there are 46 mega ball numbers and you have to match the 5 + 1. That makes the odds 1 in 175,711,536. So, with 78,130,400 tickets, the odds that SOMEONE would win the jackpot are 1 in 2.24 (175,711,536 divided by 78,130,400). However, 3 people hit the jackpot that drawing, so the way I calculated the odds of that happening was: 2.24 x 2.24 x 2.24 which came out to about 11.2. 1 in 11.2, which explains why 3 people don't normally hit the jackpot.

However, last night, there were only 22,844,680 tickets sold, and yet 34 people matched the 5 white numbers. That's 1 out of every 671,902 tickets.

I can't use the above method for figuring out what the odds of that happening are, because 3,819,816 divided by 22,844,680 says that the odds of ONE person matching the 5 numbers was 1 in 0.16, and I can't multiply 0.16 by itself 34 times to figure out what the odds of 34 winners was, because 0.16 just keeps shrinking the more you multiply it by itself.

So, given that the odds of matching 5 numbers are 1 in 3,819,816, and in yesterdays drawing 1 out of every 671,902 tickets matched the 5 numbers, what was the odds of that happening?

 

[greg555]

People often pick lucky numbers (7 or 11) or birthdays which means the numbers on tickets sold aren't totally random. So if a common lucky number is drawn or low numbers that correspond to dates (1-12 or 1-31) are drawn more people that usual will share the prize.

Greg

 

[MegaMillions]

People often pick lucky numbers (7 or 11) or birthdays which means the numbers on tickets sold aren't totally random. So if a common lucky number is drawn or low numbers that correspond to dates (1-12 or 1-31) are drawn more people that usual will share the prize.

Greg

Okay but neither of the sets of numbers that I used as examples had primarily low numbers in them.

And besides, 70% of tickets are quick picks anyway.

Even with your logic, it's bizarre that 1 out of every 671,902 tickets matched the 5 white balls. Based on the fact that there were 22,844,680 tickets in the drawing, and the odds of matching 5 white balls is 1 in 3,819,816, can't somebody figure out how to calculate what the odds of there being 34 match 5 winners are?

 

[Nickygoat]

You might want to check your calculation for the odds of someone winning from 22m tickets......

It might become a bit easier.

In my defence I haven't had any coffee yet.

 

[MegaMillions]

You might want to check your calculation for the odds of someone winning from 22m tickets......

It might become a bit easier.

In my defence I haven't had any coffee yet.

What do you mean? With 22m tickets sold, there should have been about 6 winners who matched 5 white balls. Instead there were 34. I'm wondering what the odds of that happening were.

 

[Nickygoat]

What do you mean? With 22m tickets sold, there should have been about 6 winners who matched 5 white balls. Instead there were 34. I'm wondering what the odds of that happening were.

Except in your post you did the calculation the other way round and ended up with 0.16. Unless I read that bit wrong.

Your method is sound btw but if you're using 0.16.....

 

[MegaMillions]

Except in your post you did the calculation the other way round and ended up with 0.16. Unless I read that bit wrong.

Your method is sound btw but if you're using 0.16.....

Yeah.. that's the thing. I didn't know how to continue from there because what do you do with 0.16?

By the way, here are the previous draws, to illuminate how extreme this is:

5/12/2009: 18,246,600 tickets. 4 match 5 winners.
5/8/2009: 17,744,600 tickets. 4 match 5 winners.
5/5/2009: 18,484,640 tickets. 3 match 5 winners.
5/1/2009: 78,130,400 tickets. 21 match 5 winners.

I don't think there has EVER been more than 25 match 5 winners in any drawing in the history of the game before.. let alone on a draw where there is only around 20m tickets (as opposed to a draw when the jackpot gets above 300m and nearly 100 million tickets are sold).

 

[t0mat0]

p(A Ç B) = p(A) x p(B) ?
http://richardbowles.tripod.com/maths/probability/prob.htm

p(Both A and B happening) = p(A happens) x p(B happens given that A has happened)

If outcomes A and B are independent, then the probability of B happening is not affected by whether A has happened, i.e. p(B | A) = p(B). In the special case when A and B are independent, the AND rule simplifies to the following:

p(A Ç B) = p(A) x p(B)

?

Sounds like the odds of a freak wave. Low, but they do happen.

 

[MegaMillions]

Actually, I was wrong. There were 36 winners of match 5 in the 8/31/2007 drawing, when 141 million tickets were in play.

It's absolutely insane that there were 34 winners when only 22m tickets were in play... a freak wave indeed.

But using the same means that I calculated the odds of 3 people hitting that other jackpot.. can't we calculate mathematically what the odds of this happening were?

 

[Rodimus Prime]

but like others have pointed out certain numbers people use are more likely to be picked. Lucky number/ birthdays.

Also the quick picks are not true random numbers either. A computer can not choose a true random number. It close but not quite there. You start running out a random number generator long enough it starts repeating on itself choosing the same string of random numbers as it did at the beginning.

Now the way the lotto balls are choose is a truly random event. And since we know that even the quick picks are not a true random it is possible that the winning lotto numbers happen to be a set of numbers that repeating set in a random number generator of the quick picks.

 

[MegaMillions]

but like others have pointed out certain numbers people use are more likely to be picked. Lucky number/ birthdays.

Also the quick picks are not true random numbers either. A computer can not choose a true random number. It close but not quite there. You start running out a random number generator long enough it starts repeating on itself choosing the same string of random numbers as it did at the beginning.

Now the way the lotto balls are choose is a truly random event. And since we know that even the quick picks are not a true random it is possible that the winning lotto numbers happen to be a set of numbers that repeating set in a random number generator of the quick picks.

But 34 match 5 winners out of 22 million tickets (when on average there should be 6) has never happened before and will never happen again.

I calculated that the chances of 3 people hitting the jackpot in that other drawing was 1:11.2. Using the same method, can't we calculate the chances of this? That 0.16 is the only obstacle...

 

[Rodimus Prime]

But 34 match 5 winners out of 22 million tickets (when on average there should be 6) has never happened before and will never happen again.

I calculated that the chances of 3 people hitting the jackpot in that other drawing was 1:11.2. Using the same method, can't we calculate the chances of this? That 0.16 is the only obstacle...

I just pointed out how it is possible that it went against the odds so strongly. the Quick picks if you look at them over the long term will tend to pick certain number more often than others due to the fact that a computer can not do a true random number.

 

[MegaMillions]

I just pointed out how it is possible that it went against the odds so strongly. the Quick picks if you look at them over the long term will tend to pick certain number more often than others due to the fact that a computer can not do a true random number.

There's still a number i'm looking for though.

If 50 tickets are sold for a game, and the chances of a ticket winning the top prize are 1 in 200, then the chances of 1 person winning the top prize is 1 in 4.

If 22,844,680 are sold for a game, and the chances of a ticket winning the top prize is 1 in 3,819,816, then the chances of 34 people winning the top prize is ???.

 

[gnasher729]

There's still a number i'm looking for though.

If 50 tickets are sold for a game, and the chances of a ticket winning the top prize are 1 in 200, then the chances of 1 person winning the top prize is 1 in 4.

If 22,844,680 are sold for a game, and the chances of a ticket winning the top prize is 1 in 3,819,816, then the chances of 34 people winning the top prize is ???.

It doesn't work that way. You might have some Homer Simpson type buying 34 tickets with the same numbers. Some combinations happen more often than others. I don't know how you choose your numbers in this game, do the numbers form a pattern? Are the numbers similar to those drawn in a different lottery in the previous week? If you used any system to choose the numbers and thought nobody else would use this system, it is quite likely that 34 other people used it. Say "ages of the five main actors in TV series XXX".

 

[MegaMillions]

It doesn't work that way. You might have some Homer Simpson type buying 34 tickets with the same numbers. Some combinations happen more often than others. I don't know how you choose your numbers in this game, do the numbers form a pattern? Are the numbers similar to those drawn in a different lottery in the previous week? If you used any system to choose the numbers and thought nobody else would use this system, it is quite likely that 34 other people used it. Say "ages of the five main actors in TV series XXX".

But overall things average out.

Besides, the calculation that i'm looking for WILL accurately describe how often this happens. So therefore it is correct.

Every drawing, the numbers of every prize category closely match the averages. If the odds of winning $2 are 1:75 and there were 20 million tickets sold, there will be around 266,666 winners of $2. This is just the way it works. If there were suddenly 1,500,000 winners of $2 in a 20 million ticket drawing, you could theoretically calculate the odds of that happening, and it would only happen once every that many times. (i.e. if the odds are 1:300, it will only happen about once every 300 drawings).

If we mathematically calculate the odds of 34 match 5 winners in that drawing, assuming that all numbers on all tickets are completely random, the resulting odds WILL accurately describe how often such a thing happens.