Duck race probability

[helston90]

First things first Happy Easter all.

I know a few of you love a bit of math and I've been mulling this one over all morning.

This afternoon I'm going to a charity duck race.

There are 1200 ducks, of which I have my name on 5. All of my ducks are in the 500 range.
There will be 12 races 1-100, 101-200, 201-300 etc. etc. until they have 12 finalists who then battle it for the big prize.

Part of me is gutted that all my ducks will be in the same race - therefore limiting my chances of multiple wins, however I now have 5 times more likelihood of winning a qualifier round (incl a little prize).

Would I have been better getting ducks from across the 1-1200 range rather than all in together?

 

[Markymark]

My feeling( probably not scientifically sound) is that it does not affect your chances of winning of overall. The only affects is when your ducks will get eliminated. If the prize is simply winner takes all then it makes no diffference. However if there's other prizes for heat wins etc then the rate in which you're dropping out might be affected....probably!

Ok, thought about it some more. Having multiple ducks in one category means you have a higher chance of a single small prize. Multiple categories would have meant less chance but the potential to win multiple prizes. Overall the average rate would be the same. 50:50 of winning £1000 or 25:75 of winning £2000 will on average if played enough times yield the same winnings.

Either way, just take a BBQ and some hoisin sauce then you're guaranteed to win!!

 

[nickyboy]

First things first Happy Easter all.

I know a few of you love a bit of math and I've been mulling this one over all morning.

This afternoon I'm going to a charity duck race.

There are 1200 ducks, of which I have my name on 5. All of my ducks are in the 500 range.
There will be 12 races 1-100, 101-200, 201-300 etc. etc. until they have 12 finalists who then battle it for the big prize.

Part of me is gutted that all my ducks will be in the same race - therefore limiting my chances of multiple wins, however I now have 5 times more likelihood of winning a qualifier round (incl a little prize).

Would I have been better getting ducks from across the 1-1200 range rather than all in together?

You have more chance with single ducks in separate races. The reason is that, with all the ducks in one race, it is only possible for one duck to proceed to the final. By definition, even if you win the qualifier, the other four ducks you have will lose.

If they are in separate races then even if one duck wins, the other four still have a chance to win their races . Because there are quite a lot of ducks in each qualifier the difference is small ( I can't be bothered to do the maths) but it is there

Edit....to make this a bit simpler. Imagine you have the 5 fastest ducks in the race. If they all go in the same qualifier you get one duck in the final. If they all go in separate qualifiers you get all five ducks in the final

 

[Smokin Joe]

Give the ducks a shot of EPO and they'll all win.

 

[Custom24]

Edited to say that my attempt at this was not right. You will have to read the whole thread to see why, sorry!

Thanks to @nickyboy and @swansonj for pointing out my mistakes

Whatever way you look at it, that's a hell of a lot of ducks

This is a good question. I think I have the answer right... it's quite counterintuitive. You have about 5 times greater expected gain of both the little and big prizes if your ducks are all in the same heat. It seems that at Easter, you should put your eggs in one basket. (this was incorrect - it doesn't make any difference at all)

I am assuming here the ducks are all random, there is no EPO, Lance the Duck or other shenanigans involved, and you don't have any insider knowledge on which ducks to select.

Let's call the value of the Little Prize for winning a heat L
Let's call the value of the Big overall Prize B

In the case where your five ducks are all in the same heat, you have a 5/100 chance of winning L, 5%
If you do win a heat, your duck then has a 1/12 chance of winning B, but L has to have happened first, so your overall chance of winning B is 5/100 * 1/12 = 5/1200, about 0.42%

In the opposite case, where your 5 ducks are spread among 5 heats, things are more complex.

You have to calculate the probability of exactly one of your ducks winning a heat, of exactly 2 of them, etc.

The chance of all 5 ducks winning is (1/100) ^ 5
The chance of 4 ducks exactly winning is (1/100) ^ 4 * 99/100
The chance of 3 ducks exactly winning is (1/100) ^ 3 * (99/100) ^ 2
The chance of 2 ducks exactly winning is (1/100) ^ 2 * (99/100) ^ 3
The chance of 1 duck exactly winning is (1/100) * (99/100) ^ 4

The sum of the above comes out at 0.97%, so you have an expected gain of 0.97% of the little prize. This is smaller than the 5% expected gain you had above.

If all 5 ducks do win, then you have a 5/12 chance of winning the big prize, so you multiply this by the chance of all 5 winning their heats to get the chance of winning the big prize in this case. Because the chance of all 5 winning their heats is tiny, so is the chance of winning the big prize in this case.

You then do the same thing for 4 ducks winning, etc. The sum of the probabilities of winning the big prize for the possibilities comes out at 0.082%, so you have this expected gain of the big prize, again less than the 0.42% chance in the all ducks in one heat case.

https://docs.google.com/spreadsheets/d/1-xG8Vv8UZxtQ_fZnnvq8ZwG0j2sU1q_JeJqSW39kmJ8

 

[nickyboy]

Whatever way you look at it, that's a hell of a lot of ducks

This is a good question. I think I have the answer right... it's quite counterintuitive. You have about 5 times greater expected gain of both the little and big prizes if your ducks are all in the same heat. It seems that at Easter, you should put your eggs in one basket.

I am assuming here the ducks are all random, there is no EPO, Lance the Duck or other shenanigans involved, and you don't have any insider knowledge on which ducks to select.

Let's call the value of the Little Prize for winning a heat L
Let's call the value of the Big overall Prize B

In the case where your five ducks are all in the same heat, you have a 5/100 chance of winning L, 5%
If you do win a heat, your duck then has a 1/12 chance of winning B, but L has to have happened first, so your overall chance of winning B is 5/100 * 1/12 = 5/1200, about 0.42%

In the opposite case, where your 5 ducks are spread among 5 heats, things are more complex.

You have to calculate the probability of exactly one of your ducks winning a heat, of exactly 2 of them, etc.

The chance of all 5 ducks winning is (1/100) ^ 5
The chance of 4 ducks exactly winning is (1/100) ^ 4 * 99/100
The chance of 3 ducks exactly winning is (1/100) ^ 3 * (99/100) ^ 2
The chance of 2 ducks exactly winning is (1/100) ^ 2 * (99/100) ^ 3
The chance of 1 duck exactly winning is (1/100) * (99/100) ^ 4

The sum of the above comes out at 0.97%, so you have an expected gain of 0.97% of the little prize. This is smaller than the 5% expected gain you had above.

If all 5 ducks do win, then you have a 5/12 chance of winning the big prize, so you multiply this by the chance of all 5 winning their heats to get the chance of winning the big prize in this case. Because the chance of all 5 winning their heats is tiny, so is the chance of winning the big prize in this case.

You then do the same thing for 4 ducks winning, etc. The sum of the probabilities of winning the big prize for the possibilities comes out at 0.082%, so you have this expected gain of the big prize, again less than the 0.42% chance in the all ducks in one heat case.

https://docs.google.com/spreadsheets/d/1-xG8Vv8UZxtQ_fZnnvq8ZwG0j2sU1q_JeJqSW39kmJ8

Hmmm...precisely because it seems so counterintuitive and your maths seems so robust I took a step back and had another think.

Is it not the case that your calculated probabilities of 1 duck winning is the probability of one particular duck (from your five) winning etc? I think you need to consider that there are 5 possible ways that exactly 1 duck from your 5 could win. If I'm feeling strong enough this evening I might knock a spreadsheet together

 

[Supersuperleeds]

You've not got a ducking hells chance of winning, so I wouldn't get too quacked up about it.

 

[jefmcg]

Too lazy to read the proofs.

Simplify it to 5 heats, and 5 ducks in each heat. So 25 ducks total and you have 5.

Put all the ducks in one heat, you have 100% chance of winning that heat and zero of the others. That one duck will have a 1/5 chance of winning the final.

If you put each duck in separate heats. Each duck will have a one in 5 chance of winning it's heat, and a one in 5 chance of winning the final. So each duck has a 1/25 chance of winning, but there are 5 ducks, so you add them together which gives .... 1/5 chance of one of your ducks winning. So the odds are the same in either case.

However, if there are prizes for winning a heat, then you will have a greater chance of getting that if all your ducks in the same heat. On the other hand, if you want to enjoy the trackside excitement that is rubber ducky racing, then having ducks spread across multiple events will keep the excitement going longer.

 

[nickyboy]

Too lazy to read the proofs.

Simplify it to 5 heats, and 5 ducks in each heat. So 25 ducks total and you have 5.

Put all the ducks in one heat, you have 100% chance of winning that heat and zero of the others. That one duck will have a 1/5 chance of winning the final.

If you put each duck in separate heats. Each duck will have a one in 5 chance of winning it's heat, and a one in 5 chance of winning the final. So each duck has a 1/25 chance of winning, but there are 5 ducks, so you add them together which gives .... 1/5 chance of one of your ducks winning. So the odds are the same in either case.

However, if there are prizes for winning a heat, then you will have a greater chance of getting that if all your ducks in the same heat. On the other hand, if you want to enjoy the trackside excitement that is rubber ducky racing, then having ducks spread across multiple events will keep the excitement going longer.

I shan't reply for fear of being accused of being a condescending, male mathematician

 

[jefmcg]

If I'm feeling strong enough this evening I might knock a spreadsheet together

And you will owe me an apology when you find out the chance of winning the big prize is the same in both cases.

 

[Custom24]

Hmmm...precisely because it seems so counterintuitive and your maths seems so robust I took a step back and had another think.

Is it not the case that your calculated probabilities of 1 duck winning is the probability of one particular duck (from your five) winning etc? I think you need to consider that there are 5 possible ways that exactly 1 duck from your 5 could win. If I'm feeling strong enough this evening I might knock a spreadsheet together

I am not sure. Good point

 

[jefmcg]

I am not sure. Good point

No, he's right. I wrote the same after reading your post but before reading his.

----

Look. It's actually really simple. There are 1200 possible outcomes (one for each of the 1200 ducks winning), each equally likely. 5 of them are good for you. So your chance of winning are 5/1200 or 1/240, however if you enter it.

I'd spread them out because then you have more races to cheer.

 

[PeteXXX]

According to 'management speak', you have to put all you ducks in a row...
Start from that premise and you can't go wrong!

 

[ufkacbln]

Fishing nets a few local kids and remove the opposition?

 

[Pro Tour Punditry]

Did your ducks win?