Make A Deal

The truth is far worse than the legend of Faust would imply. No deal with the devil has to be struck to ensure that fate. Without some change of course, our souls will end up in hell without ever having to trade them away to Satan.

We need for Him to save our souls, to change our destination. The problem is that we have nothing to offer Him in trade. He needs nothing from us Romans —36 , and all our attempts to appease His wrath through religious observance are futile Isaiah Certificate: TV-PG.

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Color: Color. Edit page. Add episode. The conditional probability table below shows how cases, in all of which the player initially chooses door 1, would be split up, on average, according to the location of the car and the choice of door to open by the host.

Many probability text books and articles in the field of probability theory derive the conditional probability solution through a formal application of Bayes' theorem ; among them books by Gill [51] and Henze.

This remains the case after the player has chosen door 1, by independence. According to Bayes' rule , the posterior odds on the location of the car, given that the host opens door 3, are equal to the prior odds multiplied by the Bayes factor or likelihood, which is, by definition, the probability of the new piece of information host opens door 3 under each of the hypotheses considered location of the car.

Given that the host opened door 3, the probability that the car is behind door 3 is zero, and it is twice as likely to be behind door 2 than door 1.

Richard Gill [54] analyzes the likelihood for the host to open door 3 as follows. Given that the car is not behind door 1, it is equally likely that it is behind door 2 or 3.

In words, the information which door is opened by the host door 2 or door 3? Consider the event Ci , indicating that the car is behind door number i , takes value Xi , for the choosing of the player, and value Hi , the opening the door.

Then, if the player initially selects door 1, and the host opens door 3, we prove that the conditional probability of winning by switching is:.

Going back to Nalebuff, [55] the Monty Hall problem is also much studied in the literature on game theory and decision theory , and also some popular solutions correspond to this point of view.

Vos Savant asks for a decision, not a chance. And the chance aspects of how the car is hidden and how an unchosen door is opened are unknown.

From this point of view, one has to remember that the player has two opportunities to make choices: first of all, which door to choose initially; and secondly, whether or not to switch.

Since he does not know how the car is hidden nor how the host makes choices, he may be able to make use of his first choice opportunity, as it were to neutralize the actions of the team running the quiz show, including the host.

Following Gill, [56] a strategy of contestant involves two actions: the initial choice of a door and the decision to switch or to stick which may depend on both the door initially chosen and the door to which the host offers switching.

For instance, one contestant's strategy is "choose door 1, then switch to door 2 when offered, and do not switch to door 3 when offered".

Twelve such deterministic strategies of the contestant exist. Elementary comparison of contestant's strategies shows that, for every strategy A, there is another strategy B "pick a door then switch no matter what happens" that dominates it.

For example, strategy A "pick door 1 then always stick with it" is dominated by the strategy B "pick door 1 then always switch after the host reveals a door": A wins when door 1 conceals the car, while B wins when one of the doors 2 and 3 conceals the car.

Similarly, strategy A "pick door 1 then switch to door 2 if offered , but do not switch to door 3 if offered " is dominated by strategy B "pick door 3 then always switch".

Dominance is a strong reason to seek for a solution among always-switching strategies, under fairly general assumptions on the environment in which the contestant is making decisions.

In particular, if the car is hidden by means of some randomization device — like tossing symmetric or asymmetric three-sided die — the dominance implies that a strategy maximizing the probability of winning the car will be among three always-switching strategies, namely it will be the strategy that initially picks the least likely door then switches no matter which door to switch is offered by the host.

Strategic dominance links the Monty Hall problem to the game theory. In the zero-sum game setting of Gill, [56] discarding the non-switching strategies reduces the game to the following simple variant: the host or the TV-team decides on the door to hide the car, and the contestant chooses two doors i.

The contestant wins and her opponent loses if the car is behind one of the two doors she chose. A simple way to demonstrate that a switching strategy really does win two out of three times with the standard assumptions is to simulate the game with playing cards.

The simulation can be repeated several times to simulate multiple rounds of the game. The player picks one of the three cards, then, looking at the remaining two cards the 'host' discards a goat card.

If the card remaining in the host's hand is the car card, this is recorded as a switching win; if the host is holding a goat card, the round is recorded as a staying win.

As this experiment is repeated over several rounds, the observed win rate for each strategy is likely to approximate its theoretical win probability, in line with the law of large numbers.

Repeated plays also make it clearer why switching is the better strategy. After the player picks his card, it is already determined whether switching will win the round for the player.

If this is not convincing, the simulation can be done with the entire deck. A common variant of the problem, assumed by several academic authors as the canonical problem, does not make the simplifying assumption that the host must uniformly choose the door to open, but instead that he uses some other strategy.

The confusion as to which formalization is authoritative has led to considerable acrimony, particularly because this variant makes proofs more involved without altering the optimality of the always-switch strategy for the player.

The variants are sometimes presented in succession in textbooks and articles intended to teach the basics of probability theory and game theory. A considerable number of other generalizations have also been studied.

The version of the Monty Hall problem published in Parade in did not specifically state that the host would always open another door, or always offer a choice to switch, or even never open the door revealing the car.

I personally read nearly three thousand letters out of the many additional thousands that arrived and found nearly every one insisting simply that because two options remained or an equivalent error , the chances were even.

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