APPROACH TO: What is the probability of a football game ending with a score of 14 - 0 ?

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850mph

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So, I was inspired by a Super Bowl box pool to think a bit about mathematical probability…

I have a dozen questions about the probability of particular combinations of end-game scores, but before I post these questions, I was just wondering if some probability “problems” are just too open-ended (perhaps not enough information given) to define the question in probabilistic terms…?.?

Perhaps it’s just my old FORGETFUL age, but I backed out to a 10,000foot level and thought:

Take, for instance, “What is the probability any football game ends with a score of 14 - 0?”

If we had a history of football scores and a computer, we could perhaps bootstrap the answer….

But what if we didn’t know anything about the history of end-game scores, but did know the mechanics of the scoring system (2, 3, 6, 7, 8 pts/successful play) 153 plays per game, etc…

Is there a way to approach the answer to this problem, or is it just too open ended?

Thanks.
 
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Perhaps I can simplify my question.

Let’s say I have a coin, which I intend to flip, and there are two observers.
Let’s say further, my coin is NOT a fair coin but is “tails” on each side.

Now, I show the coin to one of the observers, who looks at both sides….
I do NOT show the coin to the other observer.

Now.. before I flip the coin…..

Surely the probability of landing “tails” is DIFFERENT for each observer!

So, in essence.. if the complete set of outcomes can NOT be defined, PROBABILITIES must DEPEND on the INFORMATION known at the time a probability is calculated, or the MODEL used to compute the probability… and may be DIFFERENT for each calculator (observer.)

Bottom line: PROBABILITY without a 100% defined result space is subjective.

Comments?
 
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Does the answer to my question lie in the fact that their are TWO distinct domains of probability math?

1) OBJECTIVE Probabilities
— where the total result set may be defined apriori
— so as to be able to express probability as the mathematical result of the ratio of successes to total outcomes

2) Subjective Probabilities
 
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850mph said:
Does the answer to my question lie in the fact that their are TWO distinct domains of probability math?

1) OBJECTIVE Probabilities

2) Subjective Probabilities
Click to expand...
Without claiming to fully understand your question I get an impression you are talking about 1) math probability theory and 2) various ways of estimating probabilities in practice.
 
850mph said:
Does the answer to my question lie in the fact that their are TWO distinct domains of probability math?

1) OBJECTIVE Probabilities
— where the total result set may be defined apriori
— so as to be able to express probability as the mathematical result of the ratio of successes to total outcomes

2) Subjective Probabilities
Click to expand...
I'd say that there are
There are probably other ways to think about probability that are not included here. You can search for these terms, as I did, to learn more. Here is one article I found touching on all three:


It can be important to think about what sort of probability you are doing at a given time.Some probability questions can be answered immediately, while others (like your OP) require statistical data (so that many of us would just ignore them as being uninteresting!).
 
Sorry.. I have more to add.. but every time I hit the submit button the site crashes… so let me say thanks and I’ll bow out for now.
 
"If we had a history of football scores and a computer, we could perhaps bootstrap the answer…."

Pro Football Reference has a searchable database of every football game going back to October 10, 1920, and you can search by a particular score.

For your specific example, the final score 14-0 has occurred 77 times out of 17,665 total games played (approx. 0.436%). 2008 was the last time this score occurred.
 
850mph said:
Let’s say further, my coin is NOT a fair coin but is “tails” on each side.

Now, I show the coin to one of the observers, who looks at both sides….
I do NOT show the coin to the other observer.

Now.. before I flip the coin…..

Surely the probability of landing “tails” is DIFFERENT for each observer!
Click to expand...
Although I know what you mean, I can't agree with what you said in bold.
Whether the person knows or doesn't know that the coin has two tails doesn't matter. The probability that the coin lands on tails is 1 or 100%.
For example, If I think that when you roll a die that the chances that it lands on 6 is 1/3 doesn't change the fact that the chances that a (fair) die lands on a 6 is 1/6.
 
Steven G said:
Although I know what you mean, I can't agree with what you said in bold.
Whether the person knows or doesn't know that the coin has two tails doesn't matter. The probability that the coin lands on tails is 1 or 100%.
For example, If I think that when you roll a die that the chances that it lands on 6 is 1/3 doesn't change the fact that the chances that a (fair) die lands on a 6 is 1/6.
Click to expand...
Yes, from the COIN’S perspective (if that can be a viewpoint) the probability is 1 or 100%…..

But if WE are analyzing a human made or natural system we only have the knowledge we have at the time we are doing the probability math to calculate or model the event… the “coins” perspective is imperceptible although in a sense, it MAY be (and only MAY be) the “true” probability.

Let me wobble on a bit more…

If before I flipped the coin, if all is described as above, however I state to both parties “oh by the way, this coin may not be fair..”

I’m introducing uncertainty…. Which eliminates the frequentist method of calculating the probability…. ie: you can NOT calculate the unbiased probability if you don’t have the complete result set.

At this point you must build a probability model in your head.

So that statement changes the choice of probability “model” in the mind of the observer who has not examined the coin.

All his/her experience with coin-tossing comes into play, with the added element of chance that the coin may not be fair.

What does an observer do?

Surely there are now THREE possibilities of the probability from the uninformed observer’s perspective

1) 100% heads
2) 100% tails
3) 50%heads/50%tails

So just the introduction of uncertainty into the problem makes it even harder to determine the “true” probability, again underlining the fact that the probability calculation must be subjective according to the model chosen.
 
What I’m driving at here, is the result of trying to come up with a personal standard methodology (an expert-like system) on how to approach the calculation of MATHEMATICAL “probability” problems apriori, and possibly without further observations.

1) We must first determine what information is relevant
2) We must then determine if we have a complete result set
3) We must choose a method to compute apriori probability
4) We must determine if we can observe the problem after our initial calc
5) We must then choose a method(s) to refine the probability posteriori

Our choice of models include

1) Frequentist
2) Monte Carlo/Bootstrap
3) Bayesian

I stuck on thinking about step two above.

I’m pretty close to agreeing with the post by Dr Peterson above…. However I’m trying to think it through from 60 years of experience to come up with a way to approach any probability question which requires a numerical solution. I took stats 40 years ago in Business School so I’m relying on empirical experience at this point.
 
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850mph said:
What does an observer do?

Surely there are now THREE possibilities of the probability from the uninformed observer’s perspective

1) 100% heads
2) 100% tails
3) 50%heads/50%tails

So just the introduction of uncertainty into the problem makes it even harder to determine the “true” probability, again underlining the fact that the probability calculation must be subjective according to the model chosen.
Click to expand...
Not all unfair coins have two heads or two tails. One can weigh down a coin in such a way that the probability of getting heads is 2/3 and of getting tails is 1/3.
 
Yes! Perfect contribution Steven G…!

Yet another wrinkle… “Not all unfair coins have two heads or two tails...”

There may be a beaver on one side, or a cow, or blank, or …..??
Or maybe it’s a funny “coin” with more than two sides…??
Or……??

Underlines my thesis that without a complete, 100% understanding of the “event space” we are in an uncertain arena, where all mathematical possibilities are subjective!
 
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