Help me settle a bet

Gone Drinking

In the pub with a friend. We entered the local football draw. You pick 4 counties of 32 to try and win.

We worked out that approximately this gives you a one in eight hundred thousand chance of winning.

My friend is saying that if he buys two tickets, his odds are halfed. I'm saying no matter how many tickets you buy, your odds stay exactly the same.

Can someone confirm the latter? I know he's wrong in regards to the halfed odds anyway haha

Insane Rambling

416k to 1

MarkR

Gone Drinking wrote: »

In the pub with a friend. We entered the local football draw. You pick 4 counties of 32 to try and win.

We worked out that approximately this gives you a one in eight hundred thousand chance of winning.

My friend is saying that if he buys two tickets, his odds are halfed. I'm saying no matter how many tickets you buy, your odds stay exactly the same.

Can someone confirm the latter? I know he's wrong in regards to the halfed odds anyway haha

This should help.
http://mathforum.org/library/drmath/view/56691.html

Yakuza

Firstly, the odds of picking the correct combination are "only" 35960 to 1 if order doesn't matter (i.e. picking Dublin, Armagh, Meath and Kerry is the same as picking Dublin, Kerry, Meath and Armagh for example). --> 32! / (4! * 28!)

Secondly, of course if you buy more distinct tickets / enter more times with different combinations, your odds of winning will improve. You might have to share the prize if someone else has the same winning entry and it's possible to choose the same combination multiple times (like the Lotto) but that's a different issue (and something that's more to do with psychology rather than mathematics).

Reduce the numbers down if you don't believe me; if there were only 32 different combinations to pick from (i.e. pick one county), would your odds of having the correct combination be the same if you had bought 1 entry as if you had bought 5? Thought not...

OP, buy your mate that pint; you lost.

locum-motion

You can look at it in two ways, depending on whether you're referring to odds in the sense of 'the chance of winning' or in the sense of 'the potential return on investment'.

For example, let's continue with Yakuza's example of just picking one county out of the 32. Let's say you buy a county for €1, and the prize if your county is drawn out is €16.

You buy one county, for €1. Your chance of winning is 1 in 32 (or 31-1), and your potential return on investment is 16-fold.

Your mate buys two counties, for €2. His chance of winning is 2 in 32, or 1 in 16 (or 15-1), and his potential return on investment is 8 fold.

So, your mate has twice the chance of winning. But if he does win, he'll only get half the return that you would.

A 1 in 32 chance of a 16-fold return = 0.5.
A 1 in 16 chance of an 8-fold return = 0.5.

Whichever way you look at it, though, the overall net result is the same.

If the draw is a random lottery (which I don't think it is) then your odds are improved by having more tickets.
I.e, if it's a fair lottery/raffle with 32 unique tickets & outcomes available and you buy 1, you have a 1/32 chance of winning. If you buy 2, you have a 2/32 chance of winning ....

However, if it's not a fair lottery, then each team doesn't have the same probability of winning the underlying competition. I.e, consider a "pools" for the Premiership. Having a ticket with Hull on it is is worth less than a Man City ticket [as Man City are more likely to win the Premiership than Hull - P(M_w) > P(H_w) ]. I could own 6 tickets (QPR, Hull, Burnely, Aston Villa, Stoke & Leicester) that have less combined probability of winning than a single Chelsea ticket. Do these tickets cost the same?

Also, consider the event that tickets are not unique. In that scenario once again we have an issue. Let's consider that only yourself and one other person draws choices. And that you both choose exactly the same teams. And that those teams are then victorious. Do the rules of the lottery mean you split the prize 50/50? Now consider if you'd bought two tickets to that persons 1 (where again, all tickets have the same choices). You now split the prize in the ratio of 2:1. Depending on the ticket price, you have improved your return (though not doubled it).

In short, you cannot answer your question without more information. Different scenarios offer different answers to the question.