How To Calculate Lotto Odds? (MATH ALERT)

Factual Questions
1

Help me guys… I’m pretty good at math, but I’m stumped. We’re having a lotto-type contest with ball drop and I need to post the odds of winning. BUT, I can’t seem to calculate anything but the jackpot odds. We use numbers 1-50 and grab 6 balls. So, I figure the odds for hitting the jackpot are: 6/50 * 5/49 * 4/48 * 3/47 * 2/46 * 1/45 = odds 1:15,873,016

Is that not right? Texas Lotto has the same setup (six balls, 1-50), but they say odds of jackpot are 1:15,890,700. Are they just rounding?

Further, how do I figure the odds of matching 5 out of 6 and 4 out of 6? I tried doing 5/49 * 4/48 … 1/45. But this does not seem to turn out right and does not match up with similar odds I’ve seen. Is there another forula for figuring less-than-jackpot odds?

I’ve drained my brain trying to get these odds together (it’s the law… you gotta have 'em). Any math lovers out there? I’ll send you a clean nickel. Thanks!

Infamus Harding

2

First, you seem to have made an arithmetic error. The number the Texas lottery commission gave is correct if it was given a fraction (but not as odds). The probability of winning the jackpot is 1 / 15,890,700. Strictly speaking, the odds are 1 : 15,890,699. Odds are always given as (chances of winning) : (chances of losing).

I’ve derived a rather complicated-looking formula that I think is right for the 4-out-of-six and five-out-of-six questions you asked, but I don’t guarantee that it’s valid. It has been a number of years since I dealt with this sort of thing. You should probably consult a textbook on finite mathematics.

Let b=the number of “bad” balls (in these examples, 44)
let g=the number of “good” balls (in these example, 6)
let n=the number of balls you have to pick correctly (in these examples 6, 5, or 4)
let P be the probability of winning

P=g! g! b! b! / n! (g-n)! (b+n-g)! (g+b)!

If you have to pick 6 numbers, P=1 / 15,890,700
If you have to pick 5 numbers, P=1 / 60,192
If you have to pick 4 numbers, P=1 / 560

The formula given is more general than is really needed to answer the specific questions you asked. Nevertheless, its use is limited. It applies only to lottery-type games where the number of guesses you make equals the number of balls the lottery commission draws. (It would not apply if you are given, for example, 8 tries to guess any 4 of the 12 balls picked by the lottery commission.)

Work is the curse of the drinking classes. (Oscar Wilde)

3

*bibliophage: First, you seem to have made an arithmetic error. The number the Texas lottery commission gave is correct if it was given a fraction (but not as odds). The probability of winning the jackpot is 1 / 15,890,700. Strictly speaking, the odds are 1 : 15,890,699. Odds are always given as (chances of winning) : (chances of losing). *

That’s not how odds are given. They’re given as (chance of winning) : (all chances). E.g., the odds for winning a coin toss are 1:2, not 1:1. The latter means a sure thing.

The number of combinations of n balls taken r at a time is:
<BLOCKQUOTE><font size=“1” face=“Verdana, Arial”>code:</font><HR><pre>
n!
nCr = --------
(n-r)!r!




For the Texas lotto, that is 15,890,700.

------------------
You must unlearn what you have learned. -- Yoda
4

Wanna bet, AWB? Are you saying a 5-4 favorite has a 125% chance of winning? My dictionary defines “odds” as

A 5-4 favorite has a 5/9 chance of winning. The odds of getting heads in a fair coin toss is 1 : 1.

5

The number of possible balls you can draw for the first ball is 50. For each possibility of first choice, you have forty-nine possible second choices. For each of those 2450 possible combinations there are forty-eight possible third choices. For each of those 117600 possible sets of three numbers you have 47 possible choices of fourth ball. That gives you 5,527,200 possible positions from which you have 46 additional possible selections. At that point, with 254,251,200 possible draws, you have either 45 possible choices, or, if the lottery includes a “Magic Ball” you have another possible number, perhaps 50, or perhaps a smaller number. If it is 45, or a straight fifty choose six, you have 1,144,130,400 possible combinations. With a payoff of less than a billion dollars, it is a poor bet.

Imagine my signature begins five spaces to the right of center.

6

*Triskadecamus: The number of possible balls you can draw for the first ball is 50. For each possibility of first choice, you have forty-nine possible second choices. For each of those 2450 possible combinations there are forty-eight possible third choices. For each of those 117600 possible sets of three numbers you have 47 possible choices of fourth ball. That gives you 5,527,200 possible positions from which you have 46 additional possible selections. At that point, with 254,251,200 possible draws, you have either 45 possible choices, or, if the lottery includes a Magic Ball you have another possible number, perhaps 50, or perhaps a smaller number. If it is 45, or a straight fifty choose six, you have 1,144,130,400 possible combinations. With a payoff of less than a billion dollars, it is a poor bet. *
[list=1]
[li]Your last number should be 11,441,304,000[/li][li]What you’ve done is calculated the number of permutations of drawing 6 balls from a pool of 50. That formula is[/li]<BLOCKQUOTE><font size=“1” face=“Verdana, Arial”>code:</font><HR><pre>
n!
nPr = ------
(n-r)!



This is like saying 50*49*48*47*46*45, like you said.

Remember, though, that once they're drawn, they're sorted to give the winning combo. E.g., if 15, 20, 2, 3, 44, and 25 are drawn, the winning line is given as 2, 3, 15, 20, 25, & 44. So the initial permutation has to be divided by the number of permutations of 6 balls taken 6 at a time, 6! This is known as the combination.
<BLOCKQUOTE><font size="1" face="Verdana, Arial">code:</font><HR><pre>
         n!
nCr = --------
      (n-r)!r!

You must unlearn what you have learned. – Yoda

7

1:1 is a 100% chance of winning. A “50/50” chance (although 50 divided by 50 is 1) is a 50% chance of winning/50% chance of losing (the “/” being used as you would when you say “and/or”). So a coin toss is 1:2 odds.

If you’re paying odds, “2-to-1” would mean that you will pay double the wager if the bettor wins; not that there is a 200% chance of winning.

“I must leave this planet, if only for an hour.” – Antoine de St. Exupéry

Are you a turtle?

8

Here in Vegas, 6:5 means “6 to 5.” That means you get 6 dollars for every 5 dollars you bet, and you keep the bet. This is the correct odds bet for a 6 or 8 before a 7 in craps. The probability of rolling a 6 or 8 before a seven is 5/11.

In roulette, the payoff for black/red is 1:1. That’s 1 to 1, meaning even money. You get paid $1 for every $1 you bet, and you keep the bet. The probability of black, opposed to red (on a wheel without the greens, of course) is 1/2.

I think many are confusing, for example, “1 to 1” with “2 for 1.” “2 for 1” means you get paid $2 for every $1 you bet, but you don’t keep the bet. This is the same as “1 to 1.”

The notation is very confusing. I think Vegas likes it that way.

There’s always another beer.

9

We’re not taking any more bets. Bibliophage is a sure winner. AWB and Johnny L.A. finish out of the money. Odds are expressed as N to 1 so the 50% chance of heads when tossing a fair coin is expressed as 1-1.

10

<< With a payoff of less than a billion dollars, it is a poor bet. >>

It’s always a poor bet, the odds are still a 15.9 million-to-one against you (however you define “odds”).

Don’t confuse the mathematical Expected Value with your expected winnings. Mathematical Expected Value is the expected winnings IF YOU PLAY THE GAME BILLIONS OF TIMES (where “Billions” is used to mean some huge number of times.)

If you pay $1 for each play, for instance, then if you play this game 32 million times, costing $1 each time, you will probably win once or twice. (You could be lucky and win more than that, you could be unlucky and not win at all in those 32 million plays.) If the payoff is a billion dollars, then that’s a good return on your investment IF YOU CAN AFFORD THE $16 million to $32 million that it will probably cost to play until you win.

Of course, at one game a week, that’ll take you over 600,000 years.

Again, don’t confuse the mathematical expected value of the GAME with your real-life expectations.

11

From the “Mathematics Dictionary” (James/James), 4th edition:

And for those of you who fell asleep in math class, 1:2 is another way of writing 1/2.

You must unlearn what you have learned. – Yoda

12

But you are comparing two different concepts. 1:1 implies a direct comparison between two substances. 1/2 is usually thought of as the fraction one-half, which is used in relationship to a different (unstated in this case) substance.

For example, if you were following a chemical recipe that called for equal parts acid and water, your ratio would be written 1:1 acid to water. Equal.

Extapolate that to your coin flip. 1:1 expected ratio heads to tails. Equal chance.

The fraction 1/2 does not necessarily mean the same thing as 1:1 in this case. Following the chemical resipe you would add 1 part acid to 2 parts water. This is not a 50-50 relationship.

You would have to define your fraction 1/2 to specifically mean one part water to 2 parts total solution to make your recipe come out right. You could do this, but it is not conventional use.

So, back to the odds thing, saying 1:1 odds is equal to a 50-50 chance heads to tails, just like jcgmoi and others have stated.

13

Everybody is so concerned about correcting my “error” as to the definition of odds that you have all failed to notice that I really did make a mistake. There is a factor of (g-n)! missing from the denominator of the formula I gave above. The correct formula is:

P= g! g! b! b! / n! (g-n)! (g-n)! (b+n-g)! (g+b)!

Probability of picking 4 correctly: 1 / 1120
Probability of picking 3 correctly: 1 / 60
Probability of picking 2 correctly: 1 / 7.8
Probability of picking 1 correctly: 1/ 2.4

The figures I gave for 6 and 5 correct picks above are still correct. How can the old and new formulas give the same answer? It boils down to the fact that 0! and 1! are both equal to 1. The old answer for picking 4 was off by a factor of 2. By the way, the probabilities I give here now jibe approximately with the ratio of winners given at Lotto winners. I can’t find any official theoretical odds published by the Texas Lotto commission for any but the jackpot.

14

I think bibliophage nailed my question in the end. I never knew there was that much to getting these odds. I did fall through the same freaky odds questions that you all are talking about. And to be honest, I started out with the odds in the billions before I realized that that was impossible.

Thanks to everyone for the help. I think I’ve got plenty to go on.

Infamus

15

Maybe it’s clear to some and maybe it isn’t, but I think beeruser brought out the point that there’s a difference in odds as the ratio of winning chances to total and what is usually called “payoff odds”, which remove the amount bet from the total.

this encourages, especially the way games are played, that you “keep” the bet when you win. In fact, you win the bet back, since you don’t keep it when you lose. You don’t lose money, you simply pay to play; win or lose, that money should be considered a debit to you.

Most people tend to view track betting in this way, since you don’t see “your” money sitting there on a table; you paid for a particular ticket, and only get something if that ticket wins.

Why bother with all this? The main reason it’s important is to figure out if a bet is fair or not. You adjust the payoff odds to real odds and see if those correspond to the probability. (for example, in roulette, the payoffs are all based on 36 numbers, while most wheels have 0 and 00; this means the house gets about 5% of all the money bet on the wheel, as long is it stays fair.)

panama jack

not as jacked as i look

16

Lottery winners don’t get all the funds. A certain percentage is kept to pay for the lottery tickets, staff, schools, etc.

I think some of you forgot to take that part out.

17

Also according to biblio, odds are one in 15,890,699 first ticket. But also, the second ticket has the same odds, right? one in 15,890,699, same with the third ticket, etc. Seems more confusing, because they can all win, right?

“‘How do you know I’m mad’ said Alice.
'You must be, ’ said the Cat, ‘or you wouldn’t have come here.’”

18

This sentence is where you went wrong. With a fair coin, the probability of winning on one toss is 1/2, which is one-half, or 50%.

It’s generally accepted that when you talk about odds, it’s expressed as (prob of winning) :(prob of not winning). So the probability is 1/2, but the odds are 1:1.

Math is the one class that I didn’t fall asleep in.

19

That was supposed to be
(prob of winning):(prob of not winning) !

Damn smileys!

20

‘With a fair coin, the probability of winning on
one toss is 1/2, which is one-half, or 50%.’

You mean there can be half of one chance? :slight_smile:

trick, I like the way you figure it.
50 possible first choice, 49 the next, etc. I don’t think it matters one way or the other what the order is.