Duck race probability

[nickyboy]

With apologies to @jefmcg (who was right) here is (hopefully) the proof that the odds are the same regardless of whether you have 5 ducks in 1 heat or 1 duck in 5 heats

https://docs.google.com/spreadsheets/d/1JGcT8zm_JK1cUpZQbpw_dKUiVlLzHA5FlRGciYeEMtI/edit#gid=0

 

[Custom24]

With apologies to @jefmcg (who was right) here is (hopefully) the proof that the odds are the same regardless of whether you have 5 ducks in 1 heat or 1 duck in 5 heats

https://docs.google.com/spreadsheets/d/1JGcT8zm_JK1cUpZQbpw_dKUiVlLzHA5FlRGciYeEMtI/edit#gid=0

You need to make the sheet so that anyone with the link can view. I can't see it.

I am still not convinced (although intuition says it must be right). You are right that I was not taking into account the number of ways the heats could play out properly.

When I take into account the number of ways in which all 5 ducks can win their heats (just one way), in which 4 of the ducks can win their heats (5 ways), etc, I still don't get numbers that add up. It makes things closer, but not the same. I have updated my spreadsheet.

https://docs.google.com/spreadsheets/d/1-xG8Vv8UZxtQ_fZnnvq8ZwG0j2sU1q_JeJqSW39kmJ8/edit?usp=sharing

 

[nickyboy]

You need to make the sheet so that anyone with the link can view. I can't see it.

I am still not convinced (although intuition says it must be right). You are right that I was not taking into account the number of ways the heats could play out properly.

When I take into account the number of ways in which all 5 ducks can win their heats (just one way), in which 4 of the ducks can win their heats (5 ways), etc, I still don't get numbers that add up. It makes things closer, but not the same. I have updated my spreadsheet.

https://docs.google.com/spreadsheets/d/1-xG8Vv8UZxtQ_fZnnvq8ZwG0j2sU1q_JeJqSW39kmJ8/edit?usp=sharing

Sorry, first time I've used the google docs

https://docs.google.com/spreadsheets/d/1JGcT8zm_JK1cUpZQbpw_dKUiVlLzHA5FlRGciYeEMtI/pubhtml

Does that work?

 

[Custom24]

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.

You must be right, but I'd like to get the numbers to add up using the more complex way

 

[Custom24]

Sorry, first time I've used the google docs

https://docs.google.com/spreadsheets/d/1JGcT8zm_JK1cUpZQbpw_dKUiVlLzHA5FlRGciYeEMtI/pubhtml

Does that work?

I can't read the formulae, but I can see the results. I'm pretty sure on mine you can see the formulae. I am not sure why yours is different

 

[jefmcg]

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

You'd have been guilty of such egregious mansplaining that I would have been obliged to send @Pat "5mph" and @User13710 over to beat you up. Mansplaining is especially piquant when it's wrong.

But I want to quibble with your use of the word "mathematician". This is not the work of a mathematician.

https://docs.google.com/spreadsheets/d/1JGcT8zm_JK1cUpZQbpw_dKUiVlLzHA5FlRGciYeEMtI/pubhtml

 

[Custom24]

OK I agree that the chances of winning the Big Prize are the same. I got the numbers to add up.

But I don't agree on the little prize.

If you have all 5 ducks in one heat, you have a 5% chance of winning the little prize.

If you have 1 of each of your ducks in 5 heats, I get an expected gain of 4.901% of the little prize.

 

[nickyboy]

You'd have been guilty of such egregious mansplaining that I would have been obliged to send @Pat "5mph" and @User13710 over to beat you up. Mansplaining is especially piquant when it's wrong.

But I want to quibble with your use of the word "mathematician". This is not the work of a mathematician.

Well I knocked it up over a glass of wine while watching the snooker and making son#2's tea. And I'm not a mathematician at all, as those who have met me will undoubtedly testify. But my numbers are right....so I'm not sure what that makes me

 

[MossCommuter]

Ducking hell

This is hard

 

[jefmcg]

If you have 1 of each of your ducks in 5 heats, I get an expected gain of 4.901% of the little prize.

I think I see your error there. You are calculating the chances of winning exactly 1 heat. But the chances of winning at least one heat are in fact 5% (just add all the 1% together).

BUT if all your ducks are in one heat, you have 0% chance of winning 2 prizes, but if they are in separate heats you have 0.1% chance of winning 2 heats (1/100 * 1/100 * 10 combinations).

And of winning 3 is 1/100x1/100x1/100x10, .001%

And of winning 4 is 1/100x1/100x1/100x1/100x5 = 0.000005%

And of winning every race is .. 0.00000001%

So spreading the entries out gives you exactly the same chance of winning one prize and an infinitesimal chance of winning more than one.

 

[jefmcg]

But my numbers are right....so I'm not sure what that makes me

I really don't know. I was going to say arithmetician, but you didn't actually do any arithmetic, you just plugged it into a spread sheet. Is there a word for spreadsheet user? Accountant? Dilettante?

Mathematicians seek the clearest, most elegant solutions to a problem. What they do not do is:

using the more complex way

 

[themosquitoking]

calling @Tim Harford.

 

[jefmcg]

calling @Tim Harford.

Srsly? He'd say "Joan, you are right."

 

[themosquitoking]

Srsly? He'd say "Joan, you are right."

Hopefully he'd say "Hello" first.

 

[Welsh wheels]

Did the person with the winning ducks have a jiffy bag?