Which is more likely

[marksmeets302]

Thanks tanden, I saw my mistake after a double espresso.

I like the solution, but there's something nagging... I just wrote a program that generates a million sequences of length 1000, and checks how many times I win, and how many times Gazuty wins. This is the result:

hthwin: 485016
httwin: 489799
(the rest are ties)

Gazuty wins! His edge is small though and I wonder if statistics would say it is significant. I'm still a bit confused.

That said, since Kate Upton is involved I'd say we're both winners. Keep these puzzles coming, I love them!

[marko236]

Is it because they both start with H, which means through random you have a better chance of T coming out twice?

[Tanden]

Hi xitian

I don't want to step all over gazuty's thread but as he was going to sleep I'll see if I can explain.

He can tell me where I go wrong later!

In the imaginary game the rules would say that each player can only use each letter once but if player1 uses a letter that doesn't prevent player2 from also using it.

So as soon as there is an HTH that's a win for Marksmeets but he is not allowed to reuse the H at the end of that sequence because he has already used it.

However, Gazuty hasn't yet used that H so is already one step closer to getting HTT.

When there is an HTT that's a win for Gazuty but the last T is of no use to Marksmeets and so overall Gazuty has a slight advantage.

-------------------------------------------------------------------------------------------------------------------

To me, that makes the game seem biased but another way of looking at it is...

Scenario 1...

Write out 2 identical copies of a sequence of coin tosses
Give one copy to Gazuty and one to Marksmeets
Ask each player to circle every occurrence of their pattern
I say each player will find the same number of occurrences
(This is what the guy's who have written simulations have found)


Scenario 2...

Write out 2 identical copies of a sequence of coin tosses
Give one copy to Gazuty and one to Marksmeets
Ask each player to physically cut out each occurrence of their pattern
This time, Gazuty (HTT) will find the same number of occurrences as in Scenario 1
However, Marksmeets (HTH) will find fewer occurrences

The reason is that sometimes there will be a sequence like HTHTH

If you circle the HTH's you can circle 2 occurrences although they overlap (positions 123 and 345)

But when you have to cut out the occurrences you can only get one of them

The HTT's can never overlap each other and so you always find all occurrences

If you want to confirm this in a simulation then you will have to add a column that disallows any coin tosses that have already been used.

So if column X tests for the required pattern then cell X4 might say something like...
=if(or(X2=1,X3=1),0,{test for pattern})

[Wainwright]

https://en.wikipedia.org/wiki/Penney%27s_game

[xitian]

Hi Tanden,

Good points there. I agree it's totally about what the rules are. In the second case I also think that the rules are simply biased! Subtle difference though, and makes you realise how a small modification to something can have a big impact. That's definitely something I've learnt over the years and very applicable to trading and systems.

[Euler]

This is one of a few things I pull out as party pieces.

There are several clever things you can do to lead people to jump on what is surely a random event, but when their money is gone they can't prove in any way that it wasn't random.

I won't spoil it.

There are some really clever bits of maths and psychology you can use in random events to give yourself a better chance. I attended a charity ball recently and won a stack of prizes and then gave them back to the charity to raffle, after a half drunk speech on how life isn't as random as people often think it is and to be aware of that.

It was a slight swipe at people who invited me with some angst after learning of the nature of my business. My wife was suitably embarrassed and was pretty chuffed at myself.

[Lost The Rider]

There are some really clever bits of maths and psychology you can use in random events to give yourself a better chance

At the end of Deal Or No Deal (or any gameshow) the question of should you switch boxes being my favourite example of this.

Its a 50/50 chance.... isn't it.....

[spreadbetting]

There are some really clever bits of maths and psychology you can use in random events to give yourself a better chance

At the end of Deal Or No Deal (or any gameshow) the question of should you switch boxes being my favourite example of this.

Its a 50/50 chance.... isn't it.....

Deal or No deal isn't an example of the Monty Hall problem, Noel doesn't open a box at the end to increase your odds. Only advantage in opening the other box is the crap odds the banker offers.

[marko236]

This is one of a few things I pull out as party pieces.

There are several clever things you can do to lead people to jump on what is surely a random event, but when their money is gone they can't prove in any way that it wasn't random.

I won't spoil it.

There are some really clever bits of maths and psychology you can use in random events to give yourself a better chance. I attended a charity ball recently and won a stack of prizes and then gave them back to the charity to raffle, after a half drunk speech on how life isn't as random as people often think it is and to be aware of that.

It was a slight swipe at people who invited me with some angst after learning of the nature of my business. My wife was suitably embarrassed and was pretty chuffed at myself.

I'l take a shot at the raffle, if everyone had bought one ticket each they would have an equal chance of winning.

I think you bought an amount of tickets that cost less than half the value of the items won and you worked out how many tickets to buy to win more than 50% of the time?

[Lost The Rider]

Deal or No deal isn't an example of the Monty Hall problem, Noel doesn't open a box at the end to increase your odds. Only advantage in opening the other box is the crap odds the banker offers.

This was a trick question posed by my old stats lecturer when we were covering Monty Hall. You are correct it isn't a Monty Hall scenario despite portraying similar characteristics.

It makes no difference. In Deal Or No Deal when left with two boxes at the end there is a 50/50 chance of any of the two remaining values being in each of the boxes. The show is essentially a convoluted way of randomly picking 2 boxes from 22.

[Euler]

I'l take a shot at the raffle, if everyone had bought one ticket each they would have an equal chance of winning.

I think you bought an amount of tickets that cost less than half the value of the items won and you worked out how many tickets to buy to win more than 50% of the time?

They had an 'open' raffle so you bought tickets and then stuck them in a basket next to the prize. So the first mistake everybody made was most people went for the popular prizes. There was no discretion from people about how many tickets were in there already. People just went all in with their tickets to try and back their favourite prize.

So I went for the 'un-popular' prizes first. I then waited till the last minute when most people had put in their 'votes'.

Then I looked at each basket to figure out how many tickets I need to put in the basket to be more likely than not to win. So I then distributed tickets amoungst the pots to achieve that and with the spare tickets I picked my 'favourite' prizes.

They did a game of heads and tails as well, which I won. I did it by simply accepting it was random but betting against opponents by looking to eliminate the largest number of votes on each turn. I couldn't influence the outcome, but I could elimante the participants who didn't think it through and followed their biases en-masse.

[marko236]

I'l take a shot at the raffle, if everyone had bought one ticket each they would have an equal chance of winning.

I think you bought an amount of tickets that cost less than half the value of the items won and you worked out how many tickets to buy to win more than 50% of the time?

They had an 'open' raffle so you bought tickets and then stuck them in a basket next to the prize. So the first mistake everybody made was most people went for the popular prizes. There was no discretion from people about how many tickets were in there already. People just went all in with their tickets to try and back their favourite prize.

So I went for the 'un-popular' prizes first. I then waited till the last minute when most people had put in their 'votes'.

Then I looked at each basket to figure out how many tickets I need to put in the basket to be more likely than not to win. So I then distributed tickets amoungst the pots to achieve that and with the spare tickets I picked my 'favourite' prizes.

They did a game of heads and tails as well, which I won. I did it by simply accepting it was random but betting against opponents by looking to eliminate the largest number of votes on each turn. I couldn't influence the outcome, but I could elimante the participants who didn't think it through and followed their biases en-masse.

I would of given you more than a 50% chance of starting the first riot at a charity.

Sometimes better going with the crowd

[Euler]

I rather enjoyed shoving the condescending chairman's nose in it.

'Oh.... you're the guy who is trying to make a living gambling on horses aren't you?'

[gazuty]

The TED talk that sparked all this

https://www.ted.com/talks/peter_donnell ... anguage=en

If you want to miss all the jokes at the start then go to 3:45

[Wildly]

I'm going to sleep but I'll post up the solution in the morning unless it is solved overnight.

I'm having trouble falling asleep. Here's the solution

Kate Upton is tossing the coin

Gazuty, If you want less trouble sleeping make a slight change. Forget the coin, that just leaves Kate Upton and tossing.