I was trying to figure out my odds of winning the state lottery. Assume the following lottery:
54 balls, numbered 1-54.
Player picks 6 numbers (no duplicates).
To win the jackpot, the user must match all six numbers drawn.
No “powerball” extra ball.
The way I figured it was like this:
On the first ball, I have a 6/54 possibility of drawing one of my numbers.
Assuming I make the first one, I now have a 5/53 possibility of getting the second number.
After that, the remaining odds are 4/52, 3/51, 2/50 and 1/49. That makes the odds (if my math is correct)
If you have access to a scientific calculator ( and there are some online ones) you can also use combinatorics. By definition, nCr is the number of ways of selecting r objects from a set of n objects ( where order is immaterial).
25,827,165 = 54C6 is the number of ways of choosing 6 objects from 54. That is, it is the number of possible lottery selections. Of these, precisely 1 contains your 6 numbers. Thus the probability is 1/25,827,165 , in agreement with your answer above.
heres a silly, totally NON-mathematical way to grasp just how unlikely it is that you will win the lottery:
go to your local lotto place, and pick the following six numbers:
1-2-3-4-5-6
Then sit back and listen to everybody there laugh at you for being so stupid. Everybody “knows” that you won’t win unless you pick your favorite lucky numbers, birthdates, or whatever. I’m sure that in the whole history of lotteries, nobody has ever picked his six numbers in order, just like you count:1-2-3-4-5-6. But think about it–the machine that kicks out the numbered balls is totally random. So my silly sequence of 1-2-3-4-5-6 is no less likely than your favorite sequence based on birthdates of your pets, or whatever.
Yet,somehow, deep in your guts, you “know” that my sequence is impossible, right? The same goes for anybody else’s favorite sequence. Thats why I never play the lottery. For the same dollar, I can have a candy bar in my belly, or a castle in the air. Chocolate tastes better than air.
FWIW, I’ve heard that 1-2-3-4-5-6 and such are frequently chosen combinations. My strategy is to ensure my numbers are all above thirty-one. That way when I win, which I am almost certain to do, I don’t have to share a single penny with any of those “date-picking” types.
**chappachula **, your argumentation about 1-2-3-4-5-6 being just as likely (or unlikely) as any other combination, is totally correct, but 1 through 6 has one hugfe disadvantage: It’s played by many many other players (really. There are statistics about which combination is played how often). The probability for it to come is the same as for other combinations, but if it actually happens to come, there are much more winners with which you’ll have to share (the total sum in every class of the lottery is split up evenly among the winners), so you’ll end up with much less cash than if a patternless nonsense combination is drawn. If you want to play the lottery, play the numbers others don’t play; and since a huge percentage of picks are distributed among a relatively small number of possible combinations, there’s actually a chance your expected game value might be much larger if you find unpopular combinations.
I am personally acquainted with a couple who did, in fact, pick 1-2-3-4-5-6. Everyone except me thought they were crazy, but of course, it’s no less likely than any other combination.
When I pointed out to them that this is likely a commonly selected number, making it likely they’d have to share a jackpot 30 ways if they ever won, they switched to 14-15-16-17-18-19, figuring “who would start at 14?”
But I play the lottery. Why? It’s fun. I like gambling in small doses.
I remember a show about lotteries on the Discovery Channel, where someone said that the odds were so long, you had about the same chance of winning whether or not you actually bought a ticket!
In practical terms that’s not too far off. If this is a weekly draw, and if you buy 1 ticket per week, you can expect to win the jackpot roughly once every 500,000 years.
There is a Dilbert cartoon where Dogbert sells reduced price lottery tickets with only 1 in 25827165 less chance of winning. They were yesterdays non winning tickets.
