Probability calculation - can anyone help?

[peter]

The english lottery takes the numbers 1 to 50, and for £1 you get to choose any 6 numbers (obviousely between 1 and 50). If you get them all right when the random draw is made on the following Saturday, you win the jackpot - 40 or so % of all the money taken - and will in all likelyhood scoop up a few million pounds.

A few days ago at work I said you were more likely to be walking home and have a bag of money (of similar amount) thrown out of a car and land at your feet than achieve this correct 6 figure draw and you wouldn't even have to pay for the ticket. The guys I work with, all lottery fans, heartily dissagreed and said I was talking bollocks. At this point I realised I quite possibly was - but couldn't demonstrate it one way or the other, so this got me thinking (dangerous!).

Firstly what is the probability that I will draw 6 correct numbers at random out of the 50 possible options. This must be 'doable'. The probability of getting the first number drawn must be 6 in 50. probability of getting the first two is what? 6 in 50 X 5 in 49. I don't know how this works. What about the first three etc up to all 6. Can anyone tell me how to work out this probability.

The next bit has got to be harder - if something is a possibility (eg having a bag of money thrown out of a car at your feet) does it also have a probability that is caculable - and if it does how do you do it. what do you have to know to work ir out, and can that stuff be known.

In short guys, was I correct or was I indeed talking bollocks. (Indeed am I still talking bollocks! )

[TheFallen]

Well, I'm no mathematician - far from it - but the first part's easy.

First off, your opening statement is wrong. There are 49 numbers in the UK national lottery, not 50. But the rest of your probability calculation is correct.

So you have 6 chances in 49 of getting the first number correct, then (presuming you do), you have 5 chances in 48 of getting the second number correct, and so on.

The chances of winning the jackpot are then:

6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44, or to put it another way...

(6 x 5 x 4 x 3 x 2 x 1)
------------------------------------
(49 x 48 x 47 x 46 x 45 x 44)

which gives you

720
--------------------
10,068,347,520

Those 720 chances in over 10 billion work out to a probability of....

1 in 13,983,816

Which means that to have even a 50/50 chance of winning the UK National Lottery jackpot, given that there are two draws per week, you'd need to play a line in every draw for over 67,200 years ((13,983,816 / (52 x 2)) / 2. So make sure you eat healthily, take plenty of exercise and put loads in your pension plan, because you'll need to reach a well over-ripe old age.

In all seriousness, to increase your winning chances, if you are going to buy a lottery ticket for either the Wednesday or Saturday draw, only buy it on the day of the draw. Why? Because if you buy a ticket say 4 days in advance, there's a fourfold increased chance of your dying within those 4 days before the draw is made, and that would really rain on your parade if your ticket then went on and won. Aren't statistics cheery things?

As to calculating the possibility of someone hurling money at you randomly, I suggest that the best way to increase your odds here is to become a pole dancer, though I further suspect that this might involve you in a dramatic change in career, let alone gender preference.

[sgt.null]

can't play if you don't win or somesuch thing...

[peter]

That might be "In it to win it" Sarge (But in my case "Can't play, won't play).

But going back to the Fallen's point, why would bying a ticket in advance increase the likelyhood of your dying before the draw. I don't get this was that for real? If so - how could that work!

[TheFallen]

But going back to the Fallen's point, why would bying a ticket in advance increase the likelyhood of your dying before the draw. I don't get this was that for real? If so - how could that work!

It doesn't increase the likelihood of your dying... it just means you're more likely to be alive to enjoy your jackpot winnings. Basically, buying tickets in advance gives you a wider window of time in which you might die - and don't forget, you're getting older by the second - before the draw is made and thus causing your money to be wasted (from your point of view) on your lottery ticket... although your heirs might have a subsequent party, were your ticket to turn out to be the jackpot winner.

[Hashi Lebwohl]

When the first number is drawn, there are 49 numbers from which to choose. The probability that you chose the first number correctly is 1/49.

The probability of matching the second number will be 1/48, then 1/47 for the third number, 1/46 for the fourth number, 1/45 for the fifth, and 1/44 for the sixth number.

We will presume that the order of the numbers you chose is irrelevant since the only winning criterion is "the numbers match"; this is standard for most big lottery games. I have seen some smaller ones where the order of numbers, if you match them, will earn you a larger prize, but usually just matching them is enough.

This means that to match all 6 numbers you have a
1/49 x 1/48 x 1/47 x 1/46 x 1/45 x 1/44 =

1/10,068,347,520 chance.

I would wager that you are more likely to win that sort of lottery than to have a bag of money be thrown at your feet.

[Vraith]

1/10,068,347,520 chance.

I would wager that you are more likely to win that sort of lottery than to have a bag of money be thrown at your feet.

[/color]

I agree, but OTOH, you are literally more likely to be struck by lightning than win the lottery.

[Hashi Lebwohl]

You are correct. Lightning stikes...I want to say 50,000 people annually but don't quote me on that.

[TheFallen]

When the first number is drawn, there are 49 numbers from which to choose. The probability that you chose the first number correctly is 1/49.

The probability of matching the second number will be 1/48, then 1/47 for the third number, 1/46 for the fourth number, 1/45 for the fifth, and 1/44 for the sixth number.

We will presume that the order of the numbers you chose is irrelevant since the only winning criterion is "the numbers match"; this is standard for most big lottery games. I have seen some smaller ones where the order of numbers, if you match them, will earn you a larger prize, but usually just matching them is enough.

This means that to match all 6 numbers you have a
1/49 x 1/48 x 1/47 x 1/46 x 1/45 x 1/44 =

1/10,068,347,520 chance.

Tsk tsk, Hashi - your calculation is wrong by orders of magnitude, because - as you point out - you're not required to match the numbers in the order they are drawn. So your chance of having a correct number on the first ball drawn is 6 in 49 (then 5 in 48 for the second ball, 4 in 47 for the third ball etc etc), as per my example above. Your odds are therefore 1 in near on 14 million, rather than 1 in over 10 billion. Were the order of the numbers to matter, then your 10 billion to 1 odds would be correct.

Still a tad unlikely, though.

[Hashi Lebwohl]

hrm...let me think for a second...

Lo and behold you are correct. You have a pool of sic numbers, so you have a 6/49 chance of matching the first number drawn.

Statistics was never my strong subject--it just didn't make sense to me. I fought for my C and was glad for it.

[I'm Murrin]

Yeah, permutations versus combinations...


If you include the other variations on the winnings (the 5, 5+bonus, 4, and 3 number matches), what I've read on the subject suggests the average payout over many purchases makes a ticket on the UK lottery worth roughly 1/3rd of the ticket price.

(I've also seen math showing that if you buy the full 13.98 million variations on the tickets, you'll actually make a net profit - so long as noone else gets the jackpot that week.)

The highest value tickets are actually on the smaller games like Daily Play - the payoff is much smaller, but the chance of winning is high enough that you're much better off on those games, if you must play something.

Apparently the US lottery has ridiculously low odds.

[Vraith]

Apparently the US lottery has ridiculously low odds.

IIRC my state lottery is something between 1 in 40 or 50 million.
A big multi-state one [megamillions? megaball? something mega] is somewhere around 1 in 175 or 180 million.

[sgt.null]

but in texas if you get three numbers out of six you 3 dollars. more money for four or five numbers. all six gets you the jackpot.

[I'm Murrin]

Same in the UK, except it's £10 for 3, and the odds are higher .

[peter]

As to calculating the possibility of someone hurling money at you randomly, I suggest that the best way to increase your odds here is to become a pole dancer

Now that's not as bad an idea as it might sound. I have been contemplating a career change for the last thirty years or so and it strikes me that any quality pole dancing joint eager to present it's pc qualifications could demonstrate in one fell swoop (by employing me) that it is neither sexist, ageist or fatist in it's approach to staff procurement - and I bet I could put on a damn good comedy pole dancing routine at the start of the night as well!

(one point - everybody so far has gone for the (in comparison) easy option of dealing with the national lottery computation - but where is the Lancelot amongst you who will step forward and tackle the much thornier problem of how you would (if you could) calculate the probability of having said bag of money chucked at your feet).

[TheFallen]

Speaking of probability, have you guys picked over the Monty Haul conundrum? You know, the final of the gameshow, three closed doors, 2 goats and 1 sportscar? It's the one that drives amateur statisticians crazy...

I bet you have - let me know if not and I'll post it. Ever so simple but hard to explain why.

[I'm Murrin]

I know it, and it does seem counter-intuitive at first, doesn't it?

Essentially, you had a 66% chance of being wrong when you made your decision on the three doors, and removing one of the doors doesn't change the fact that there's a 66% chance you were wrong.

[Vraith]

I know it, and it does seem counter-intuitive at first, doesn't it?

Essentially, you had a 66% chance of being wrong when you made your decision on the three doors, and removing one of the doors doesn't change the fact that there's a 66% chance you were wrong.

I remember being shown a version of that...from that woman, who's supposedly the smartest woman [maybe person?] in the world trying to explain it in her column. It really is a very strange thing when you see it...my first thought was "that can't possibly be right."

[aliantha]

The only coherent thing I can add to this discussion is that it's Monty Hall, not Monty Haul. He was the emcee of a game show called "The Price Is Right".

"Susannah Jones, come on down! YOU are the next contestant on THE PRICE IS RIGHT!" <cue theme music>

Usually Susannah would be dressed as a chicken or something. It was quite the spectacle.

Ah, my misspent youth...

[peter]

Speaking of probability, have you guys picked over the Monty Haul conundrum? You know, the final of the gameshow, three closed doors, 2 goats and 1 sportscar? It's the one that drives amateur statisticians crazy...

I bet you have - let me know if not and I'll post it. Ever so simple but hard to explain why.

It does ring a bell the Fallen, but I have long forgotten the details. Could you post it if it isn't too difficult. I seem to remember it was very interesting.