The Lottery Game

tuf-web_co_uk

may have been asked about before, but i was wondering how i could find out how many possible combinations there is for the lottery.

not exactly gonna go out and buy that amount of tickets, but its just a random query.

thought about using rand() but i wouldnt know where to start with limiting the results, and processing duplicates.

if anyone knows where to start it would be great.

lottery numbers are 1 - 49

thanks, Jon

goldbug

If i'm correct, the "easy" (read: stupid) answer just requires some simple multiplication.

in this case, if all number slots had 49 possibilities, your combinations would at most be 49^n, where n is the number of balls/slots. This of course doesn't eliminate dupes, a formula for which goes well beyond my primitive math training (ug, 1 + 1 = 3)

If some of the number slots had different maximums (like many US games) it'd be different:
Say slots 1-5 had max of 50, 6th slot max of 30

Combinations = 505050505030
(or 50⁵ 30¹⁾

Another thing to take into account is how the balls/numbers are delivered. For example, some of the lottery games in Ohio, for example, use individual machines for each slot (Pick-3 or 4), or one big tank to dispense all numbers (Super Lotto), in which case the number of possibilities is less (combinations vs. permutations?). For example, individual tanks would yeild xⁿ possibilities (before dupes are removed), where one big tank (no duplicate numbers) would be x * (x-1..n) possibilities -- x is maximum number, n is number of balls

.... Where's weedpacket? He's a math man. Anything other than simple addition makes my head hurt. 🙂

drawmack

the formula is actually very simple.

It is the product of the number of entries for each slot.

For example PA dialy number uses three balls each with the available numbers being 0 - 9.

So all combinations is 10 10 10 = 1000

Pennsylvania Cash 5 uses 1 - 39 and 5 balls are selected but a given number cannot be repeated so the total available combinations are 39 38 37 36 35 = 690,908,40

If what you want to know is how to calculate these totals dynamically for different games it is pretty simple.

Step 1) Calculate the range of numbers by taking (high - low) + 1.

Step 2) If you have a static range (i.e. PA daily number) proceed to step 3, else proceed to step 4.

Step 3)

<?php
$total = $range;
for($i=1;$i<$balls;$i++)
    $total *= $range;
?>

Proceed to step 5

Step 4)

<?php
$total = $range;
for($i=1;$i<$balls;$i++)
    $total = $total * ($range - $i);
?>

Step 5)

<?php
echo 'There are ' . $total . ' combinations for this lottery.';
?>

Two more exercised remain that are left to the reader.

Exercise 1: PA Daily number allows you to play boxed and straight. The above code calculates the number of straight possibilies (i.e. 123 != 321) however it does not calculate the number of boxed possibilities (i.e. 123 == 321).

Exercise 2: Some lotteries choose some balls from one pool and some balls from another pool. This means that their range schema does not fit into either of the above coded categories.

tuf-web_co_uk

wow, alot more simple than i thought. guess its good that i'm taking a/s level maths in college!! (definately need a brush up on my maths)

anyway, i put together a selection of numbers with drawmacks code and it works fine.

the english lottery works like any other lottery, choses 7 numbers from a range of 1-49, however the 7th number is a "bonus" ball.

either way this is what i used in the end.

<?php

$ball = array(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,
21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,
42,43,44,45,46,47,48,49);

$range = max($ball) - min($ball);
$range++;
$balls = 7;

$total = $range; 
for($i=1;$i<$balls;$i++) 
    $total = $total * ($range - $i);

echo 'There are ' . $total . ' combinations for this lottery.'; 
?>

thanks again!!

Jon

The_Chancer

I think the total permutations are somewhere close to the 16million mark....

(Remember hearing it somewhere about the UK lottery....)

piersk

Are you sure it was that high? I thought it was 13 million...

The_Chancer

Well - whichever it was - I think it is less than the figure given by tuf-web's script... that is in the billions... (432938943360)

Can't research from work though - darn internet police.....

Ahh got it...

7 balls (including the bonus ball) - does mean 49484746454443 - and does come out to 432,938,943,360 Much higher than I had thought....

Oh well.... 432 billion..... seems like a shot in the dark to play that damn thing....

But if there are that many permutations, how come someone usually wins the jackpot every time.

Do we have to take into consideration that only six numbers are taken - as the bonus ball then doesn't count ?

This would then make it about 100 billion - still making the probability of this very high.

So does that mean then that there is a "lady luck" helping people win ?

Weedpacket

Originally posted by The Chancer
So does that mean then that there is a "lady luck" helping people win ?

It's hard to say without knowing how many games are played (you tend to hear more about the winners, so there's an observational bias). Here's a bit of a clue - where does the prize money come from?

piersk

Originally posted by The Chancer
So does that mean then that there is a "lady luck" helping people win ?

Shhhh... you don't mention her by that name. Just "The Lady"...

goldbug

Originally posted by tuf-web.co.uk
wow, alot more simple than i thought. guess its good that i'm taking a/s level maths in college!! (definately need a brush up on my maths)

anyway, i put together a selection of numbers with drawmacks code and it works fine.

the english lottery works like any other lottery, choses 7 numbers from a range of 1-49, however the 7th number is a "bonus" ball.

either way this is what i used in the end.
<?php

$ball = array(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,
21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,
42,43,44,45,46,47,48,49);

$range = max($ball) - min($ball);
$range++;
$balls = 7;

$total = $range; 
for($i=1;$i<$balls;$i++) 
    $total = $total * ($range - $i);

echo 'There are ' . $total . ' combinations for this lottery.'; 
?>
thanks again!!

Jon [/B]

wouldn't it just be easier to use:

$balls = 7;
$ballmax = 49;
echo 'There are ' . pow($ballmax,$balls) . ' combinations for this lottery.';

.... like I had previously mentioned...?... this yeilds combinations...

max (max - 1..n) is permutations 🙂.... which is cleaner as:

function factorial($num) {
  for ($i = ($num - 1); $i > 0; $i--) {
    $num *= $i;
  }
  return $num;
}

$balls = 7;
$ballmax = 49;
echo 'There are ' . (factorial($ballmax)/(factorial($ballmax - $balls)) . ' permutations for this lottery.';

(or use the gmp_fact() function if gmp is installed/enabled)

drawmack

but it's not a true factorial cause they're not choosing all 49 balls.

drawmack

You could do this too:

<?php

$ball = range(1,49);
$range = count($ball);
$balls = 7;

$total = $range; 
for($i=1;$i<$balls;$i++) 
    $total = $total * ($range - $i);

echo 'There are ' . $total . ' combinations for this lottery.'; 
?>

goldbug

Originally posted by drawmack
but it's not a true factorial cause they're not choosing all 49 balls.

I know... thats why you divide by the factorial(max - number).

drawmack

Originally posted by goldbug
I know... thats why you divide by the factorial(max - number).

That's a whole lot of extra processing on the cpu.

Let's see multiplication takes 5 (i think) clock cycles. You're doing 2 42 extra multiplications. That's 5 84 extra clock cycles. 420 extra clock cycles, everytime this script runs.

goldbug

Originally posted by drawmack
That's a whole lot of extra processing on the cpu.

Let's see multiplication takes 5 (i think) clock cycles. You're doing 2 42 extra multiplications. That's 5 84 extra clock cycles. 420 extra clock cycles, everytime this script runs.

I never said it was faster. But i'd wager using the proper gmp_fact() function is speedy enough.

Speaking of which... why doesn't PHP have a built-in factorial function? It seems like a "core" enough math function to include.

The_Chancer

Originally posted by Weedpacket
It's hard to say without knowing how many games are played (you tend to hear more about the winners, so there's an observational bias). Here's a bit of a clue - where does the prize money come from?

The prize money comes from the number of people who play.. £1 per play.

Average jackpot is about £8million but they do take for good causes as well as all the other smaller prizes.

Still - definately less than 100 Billion (1000 million) as that WOULD be a fantastic win....

So - "The Lady" MUST have something to do with it....

piersk

Originally posted by The Chancer
So - "The Lady" MUST have something to do with it....

Thats better 🙂

Weedpacket

Of course, all this argument is assuming that the order in which the balls are drawn is relevant. But I'm presuming that 23,45,10,8,3,19 is just as good as 3,19,8,10,23,25. So you've got a huge overestimate; you're out by a factor equal to the number of possible permutations of six balls.

The number of possible permutations of six balls drawn from a set of forty nine is 49!/(49-6)!, but the number of possible combinations is only 49!/(6!(49-6)!).

function factorial($n)
{
	static $fact=array(1);
	if(isset($fact[$n])) return $fact[$n];
	return $fact[$n] = bcmul(factorial($n-1),$n);
}
echo bcdiv(factorial(49), bcmul(factorial(6), factorial(49-6)));

drawmack

Originally posted by Weedpacket
Of course, all this argument is assuming that the order in which the balls are drawn is relevant. But I'm presuming that 23,45,10,8,3,19 is just as good as 3,19,8,10,23,25. So you've got a huge overestimate; you're out by a factor equal to the number of possible permutations of six balls.

I did mention that limitation on my initial post of my code. When the person asking the question used my code I assumed order was relevent as being from the states I have no idea how the english lotery works.

tuf-web_co_uk

7 numbers are chosen from the machine (6 normal and 1 a bonus, both chosen from the same machine)

however you can only choose 6 numbers on your ticket.

in my earlier code i forgot to change $balls=6; but even so that brings back

10,068,347,520

and as someone said before i dont think 10billion tickets are bought