# The Naked Scientists Forum

### Author Topic: Really Tough Probabilty Question Need Help!  (Read 1760 times)

#### uksceptic

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##### Really Tough Probabilty Question Need Help!
« on: 16/06/2009 17:25:16 »
Hello All,

This is my first time posting on here although I have been listening to the Naked Scientist for sometime. My question is this;

72 football teams enter the first round of the League cup which has been going for 49 years. In that time what is the probability that ANY two teams will be drawn each other three years in a row?

This question arose out of my friends and I all being Southend United supports and Southend being drawn Cheltenham in the first round of the cup for the third time this season! The discussion arose about how unlikely this was to happen by chance, being a guest writer for the Southend United fanzine I thought the discussion could make an interesting article. Attempts were made to calculate the probability of Southend drawing Cheltenham three times in a row, we worked out that the probability of this exact event happening three times in a row is 1/357911. But I argued that this was a misrepresentation of the probability since it did not consider all the other years or teams that didn't draw each other.

My understanding is that we need to work out the probability of ANY two teams meeting each other three times in a row not just the probability of Southend meeting one particular team in a row to truly understand the probability of this happening. I think it is a variation on the birthday problem.

Unfortunately whilst I can understand the problem I am not accomplished at the math behind it, the best I can come up with so far is the probability of one particular team meeting meeting ANY team 3 times in a row which is 1/5041

It would be fantastic if anyone could offer any help with this problem for a piece I am putting together for the Southend United fanzine entitled All At Sea.

Many Thanks

Martin
uksceptic

#### Bored chemist

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##### Really Tough Probabilty Question Need Help!
« Reply #1 on: 16/06/2009 19:06:20 »
Just consider Southend in the last 3 years.
In the first year they were drawn against someone- it doen't matter who.
In the second year the odds are 1 in 72 that they would be drawn against the same team. Ditto for the third year. (Assuming that nobody leaves or enters the league, have fun correcting for that).
So the odds for "three in a row" are 1 in 72*72 = 1 in 5185
That looks like quite a coincidence.
On the other hand there are equally dedicated writers for all the teams.
If, rather that Southend, it has been Example United who had been drawn 3 times in a row against  Madeupname city ten we would have heard from them instead of you.
So there are 72 teams who, if they had a coincidence like this, might have written in. That drops the odds by 72 fold (I think).

So the odds are just 1 in 72. Not that spectacular.

Incidentally, the real answer is that since it has happened, the probability is exactly 1

#### uksceptic

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##### Really Tough Probabilty Question Need Help!
« Reply #2 on: 17/06/2009 11:53:05 »
Thanks for the answer. Is this correct though? I thought it was more complicated than this!

#### DoctorBeaver

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##### Really Tough Probabilty Question Need Help!
« Reply #3 on: 28/06/2009 18:22:27 »
This is the same sort of thing as lottery numbers. The probability of getting the same 6 numbers on 2 consecutive weeks is exactly the same as getting those 6 numbers the first time round. The reason being that you start from scratch each week.

#### The Naked Scientists Forum

##### Really Tough Probabilty Question Need Help!
« Reply #3 on: 28/06/2009 18:22:27 »