In it, 3 doors are available, 2 containing goats (no prize), and 1 containing a prize. The individual selecting the door will choose one of 3. Monte Hall will then always show a goat among the two remaining doors that the player did not choose. The player can then switch doors if he wants.
Many people perceive the problem incorrectly and believe the odds are 50/50. However, the odds if the player stays put is 1/3. The odds if he switches is 2/3. The information provided as a result provides an edge, but only if you adapt correctly to this new information.
A lot of people don't understand this problem, because it can be confusing, but the key variable is that Monte Hall will ALWAYS show a goat of the 2. This means that 2/3rds of the time you actually choose wrong initially, and 2/3rds of the time after Hall has removed the goat, switching will always produce the prize. 1/3rd of the time you will choose correctly to begin with, and Hall will remove either of the two goats, and switching will fail.
This example in the movie seems to have nothing to do with blackjack, and nothing to do with football, but in fact, it has everything to do with both as well as virtually any game when there is both information and decisions.
The teacher recruited the student to his MIT blackjack team because he knew the importance of adapting to new information. This idea of adapting to new information allows card counting to produce a profit. There are 52 cards in a deck, but if you operate under the assumption of random dealing, you have no edge. However, as soon as the dealer begins dealing cards and the player adapts to the new information that is available, the player can gain an edge. (At least given the blackjack rules available at the time when the MIT blackjack team profited). Even with 4 decks of cards, if you have seen several low cards discarded, the payout of blackjack being disproportional as well as the increased probability of the dealer busting provides an advantage when lots of 2,3,4,5 and 6 are discarded (Because the dealer has to hit on 14,15 and 16, while the player can recognize the chances of busting are higher due to the new information).
Now in football, much has been made about baseline rates, success rates running or passing, and how the decisions in football are "sub optimal" according to these baseline rates.
However, every play provides new information, and using baseline rates fails to take into account variable change.
For example, surprise onside kicks succeed a very large percentage of the time that suggests a team should onside kick much, much more often, and even attempt to do so every time. What the "always onside kick" theorists are missing, is that the baseline stats are skewed by decisions that happen as a RESULT of adapting to variable change. In the case of onside kicks, a coach may see the sloppy, undisicplined player leaving before the ball is kicked and completely turning his head around. The special teams coach may be more aggressive than the regular coach who makes the final call and has the additional challenge of optics and what looks bad to fans and to a team which creates pressures on his job. But the special team coach will likely only convince the coach if it's an overwelming advantage.
I believe that head coaches are probably close to optimal in terms of keeping their job, but sub optimal in terms of running the onside kick (a player leaving early 70% of the time or turning his head partially around with bad hands may be enough information to provide an edge). However, I believe that the assumptions built in are terrible. The idea that coaches are just kicking onside randomly, that edge doesn't change as a result of information, or that it doesn't drop off dramatically when teams have even seen it on tape just a few times shows a lack of understanding of real football, and an over emphasis on theory independent of reality or without the depth of understanding why decisions are made and how information creates change.
The same thing is true on 4th down plays. Coaches are capitalizing off of variable change (although not nearly enough). In other words, the 4th down conversion rate will be slightly inflated due to the fact that teams may just try to draw a team offsides, with a word or two they can say to actually run the play, as well as the center will hike the ball if he sees a player jump offsides which will give them a free attempt at moving the chains as well as the yardage if the play fails. Teams may not intend on actually going for it, determining before the play that it's worth either taking a time out or a delay of game in order to attempt to draw opponent offsides.
This goes for fake punts as well. In high school we had a call "white" that meant punt unless a gunner was uncovered, "blue" that meant always punt" and "black" that meant always run a particular fake and "green" which was a swinging gate formation that had two separate playcall options that I won't discuss here.
I believe teams would be well served creating several formational shifts to try to get a guy uncovered or a clear mismatched, and if they get that mismatch to actually make a call then hike the ball (on 2 by default), otherwise they would not make the call and by default no play would be run. These formational shifts would be like playing blackjack with a free surrender option that cost nothing, you could essentially only bet money if you were dealt a favorable hand, otherwise take a time out or delay of game, then punt, or kick a fiedlgoal, or try it again.
The exact strategy may require some estimation of conversion rates given certain situations and down and distances, as well as how teams adjust after they see it the first time. You don't have to complicate things and remember every positive situation, just as in the hypothetical blackjack you certainly know that if you are dealt a 9 10 or 11 or many soft hands that you would play without worrying about opponent's hand. You could chose to hike the ball in situations that are determined overwhelmingly positive so you don't have to determine that on 4th and 3 only hike in these situations, but not on 4th and 4.
Formationally I would have a different plan and personel depending on down and distance. I would probably have 2 or 3 tight ends (including H-backs), a runningback and 1 or two receivers (not including athletic tight ends). I would start with 3 tight ends spread out wide right. If the B gaps are open with no LBer behind to fill, I may run the ball in the B gap (this is a play that depending on the formation and down and distance could be ignored). Otherwise, I put the runningback in motion left to a 5 wide set. If the A gaps are both open, I'd run a QB sneak (if I have 1 yard or less, maybe 2). If a WR or TE is uncovered at any time, I'd probably throw a smoke screen or drag route or quick hitch over the line depending on what was the highest probability play. If the A and B gaps are jammed up by DL and WRs are covered, after I run the runningback back in motion, I run him back in motion and give him the ball on an end around or what is called a "jet" play.
The idea is not to focus on the specifics of the formation, but that certain matchups are more favoarable and that by recognizing these situations, information can change whether going for it is correct or not.
There may be on a given week a particular mismatch. The read would be more based on personnel.
Perhaps I line up my tight ends out wide hoping the defense lines up their cornerbacks on them. so that I have a speed advantage on the inside with the WR and RB.
We could have the RB act as the FB, and WR as the RB and create two quick motions. with the extra space from WRs and TEs spread out wide providing room to run.
Or off a similar presnap scheme to switch it up and not be predictable, we can have the same look and run it outside left.
The point is, as information changes, matchup sand proabbility changes. Failing to take advantage of that is a huge disadvantage. Capitalizing off of that may provide tremendous gains for your offense and win probability..
This is basically just "one scheme" trying to rely on conventional wisdom of having big guys vs big guys and then spreading the team out instead of tradigional power run with TEs as decoys in a lot of plays or potentially as blockers on screen plays.
If the defense would have come out with a different alignment, they probably would have left an A gap open or a B gap open. There are many combinations a team would have to prepare for, but this is just an example of some formational shifts to create some advantage.
The QB could also go out of the shotgun here to provide more room to buy time and let WRs and RBs outrun linebackers.
The defense may do a good job of alignment to prevent any clear easy pickup, but still we can try to assess probability based upon attributes, alignment, etc and ahead of time determine what situations we'd go for it.
I meant for this to be more 4th and short since I have QB sneak and run up the B gap open as well as the outside C gap runs. In 4th and 3, 4, 5, 6, There would probably be different personnel and different formational options and a lot more passing options and combinations than just a quick smoke screen. Slants and in routes, flat routes, corner routs, smash concepts, scissors combination routes and so on.
Another situation that may play out is when a key defensive player gets injured prior to a touchdown and you opt to go for two knowing your odds have likely gone up.
The game of football is highly sensitive to information changing, and for anyone to base a model that neglects the massive impact of these changes doesn't really understand the game of football.
If you want to tell me Adrian Peterson trying to get yards facing 9 man fronts where there is one more defender than you have blockers is equal in value to the same number of yards gained by CJ spiller in open space, and all runs are created equal and only down and distance causes the expectations to change, then you are missing over half of what football is about. Audibles, matchups, checkdowns. The QB you have will have to do different things based upon formations and such as wel. If a team has 9 in the box, inside crossing routes and flood concepts and the deep ball off play action should be more effective if you have a guy that can beat one-on-one matchups and a QB to throw it.
If you instead open up the offense and look downfield and stretch the field horizontally with throws, there is more stress on the QB to be able to make difficult NFL throws down the seem or on a deep out in tight coverage as well as read the entire field and find the open guy. Because of all this pressure on the outside throws and the QB, it challenges the defense in a way that opens up the running room and puts a linebacker on the runningback.
A QB does not only go to a primary receiver and just change who that receiver is. He actually reads defenses and goes through progressions and reads defenders and relies on route concepts.
With a smash concept, he may read the safety and corner on the side of the field where he feels there is an athletic advantage to his players. Anyone that attempts to come up with strategy without understanding that football is a game of contingencies and changing information and adapting to new information is simplifying things to the point where it's not even modeling the game itself.
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