Football Game

[le_boucher]

Hello. If someone can explain me the solution of this situation please tell-me !

Two football teams are playing a football game. Team A is by far better than team B (let's say New England Patriots against a college Team). Score starts at 10 000 to 0 for New Engalnd.

True or False:
If they play with no time (they play continuously), it is 100% sure that College Team will have the lead at one moment.

Thanks for your help.

Le_Boucher

[Stevedye56]

Well what do you think and whats your reasoning? Just because New England is down 10,000 doesnt mean that they cant score any points and that the college team will not score any more.

[le_boucher]

New England is not down 10,000-0. They lead 10,000-0.

I'd like to know that people think before telling what's my point of view. Some friends and I don't agree since 1 or 2 years about this situation so I hope people here will help me.

[Stevedye56]

Score starts at 10 000 to 0 for New Engalnd.

I found this unclear and read it as New England has 0 its not my fault its not clear...

Some friends and I don't agree since 1 or 2 years about this situation so I hope people here will help me.

????????????????????

P.S i never told you what your opinion was so i dont understand where this is coming from."I'd like to know that people think before telling what's my point of view."

All i know is that it's not going to help you get responses

[shmoe]

Until you give some more details on what exactly "Team A is by far better than team B" means for this problem, any answer is just going to be baseless speculation.

If you can formulate it something like "every <insert arbitrary chunk of time>, team A has a probability of p of scoring a point, team B has a probability q of scoring a point and there's a probability 1-p-q of no scoring" then you can get something going. As it is, it's too vague for anything resembling a sensible analysis.

It would be a good idea to explain what you mean by "100% sure" as well. Do you mean this will happen with probability 1? Or will it *always* happen? Or what?

[pete5383]

how about this: let's say f is a number which represents the number of touch downs scored by New England (x) divided by the number of touch dows scores by the college team (y). So,

f = x/y

When f < 1, the college team is winning. When f > 1, New England is winning. So let's say New England scores 100 times as much as the college team. so x/100 = y. so then

f = 100x / x

This shows, at least to me, that f never -has- to be less than zero. Also, since it seems like you were talking about if they played forever, if you take the limit of that function as it approaches infinity, it shows that New England has 100 times as many points as the college team. So, it seems to me that theres not reason that the college team would have to lead (although, there are many people much smarter than me on the forum, so you better listen to them).

Of course, this is if New England scores 100 times as much as the college team. If this is the case, I don't see why the gap couldn't just keep getting larger.

[CRGreathouse]

I'm going to try to give this a rigorous interpretation, le_boucher. Tell me if this seems to describe what you're asking.

Since the time is unbounded, we can just compare the chance that the good team P scores compared to the poor team C. I'm going to assume that each point is independent. Call P's probability of scoring the next point p, and give C probability 1-p. Let P(n) be the number of points the good team has after n points have been scored, and C(n) the number of points the bad team has after n points have been scored. Let P(0) = 0 = C(0) and P(n) + C(n) = n.

I'm interpreting your question thus: For all p < 1, is it true with probability 1 that P(n) + 10000 < C(n) for some n > 0?

[le_boucher]

Thanks for yours anwsers and by the way sorry for my english.

CRGreayhouse you are right in your interpretation of my question. It's very clear like that. Thanks. Ill be waiting for your comments.

[shmoe]

With that interpretation, this is the standard "gambler's ruin" problem. You can think of team P as starting wtih 10,001 (I'll explain why in a moment). At each play team P has probability p of gaining a point, and probability 1-p of losing a point. Note that team P losing a point is equivalent to team C gaining a point.

The gambler's ruin problem will give the probability that team P ends up at 0 some point in an infinite sequence of trials. Team P ending up at 0 is equivalent to team C taking the lead. This is why we start P with 10,001 points in this formulation, if we started at 10,000 gamblers ruin would just give the probability of a tie.

We assume p>1/2 since they are the better team. In this case, the probability P eventually hits zero is given by ((1-p)/p)^10001. Note this is strictly less than 1, so it is not a 'sure thing' that the poorer team will catch up.

see http://www.columbia.edu/~ks20/4700-05/4700-05-Notes-GR.pdf#search=%22gamblers%20ruin%22

it's equation (4) in there you are interested in.

[Stevedye56]

Until you give some more details on what exactly "Team A is by far better than team B" means for this problem, any answer is just going to be baseless speculation.

If you can formulate it something like "every <insert arbitrary chunk of time>, team A has a probability of p of scoring a point, team B has a probability q of scoring a point and there's a probability 1-p-q of no scoring" then you can get something going. As it is, it's too vague for anything resembling a sensible analysis.

It would be a good idea to explain what you mean by "100% sure" as well. Do you mean this will happen with probability 1? Or will it *always* happen? Or what?

Glad to know i wasnt the only confused...

[CRGreathouse]

We assume p>1/2 since they are the better team. In this case, the probability P eventually hits zero is given by ((1-p)/p)^10001. Note this is strictly less than 1, so it is not a 'sure thing' that the poorer team will catch up.

see http://www.columbia.edu/~ks20/4700-05/4700-05-Notes-GR.pdf#search=%22gamblers%20ruin%22

it's equation (4) in there you are interested in.

If the better team has a 50.01% chance of winning each game, then the chance that the worse team will ever be ahead is < 2%. If the better team has a 50.02% chance of winning, the worse team has a < 0.034% chance of ever being ahead.

[AKG]

No, it's not 100% sure that the College Team will have the lead at some point. It is entirely possible that both teams never score.

[le_boucher]

All of you guys Shmoe, Stevedye56 and CRGreathouse agree and "The Gambler's Ruin Problem" agree too that if odds are better than 1/2 for the Great Team, there is a positive probability that Poor Team will never come back.

THANKS to all of you and see you soon!
le_boucher