Marylin Vos Savant & the Goats: A New Look
Introduction: In a previous lecture, we gave a New Definition of Probability. In this lecture, we will show how this definition enables us to resolve the massive amount of controversy which surrounds an apparently simple probability puzzle, known as the Monty-Hall problem. The book The Power of Logical Thinking, quotes cognitive psychologist Massimo Piattelli Palmarini: “No other statistical puzzle comes so close to fooling all the people all the time [and] even Nobel physicists systematically give the wrong answer, and that they insist on it, and they are ready to berate in print those who propose the right answer.”
Before proceeding to discuss this problem, we review the key features of our new definition:
1. Probability is a feature of external reality, not an aspect of our ignorance/knowledge.
2. Probability exists PRIOR to occurrence of random event.
3. Probability is EXTINGUISHED after event occurs.
We will use this definition to provide a new answer to the Monty Hall Problem. Our definition gives new insights — both a new solution and an explanation of why the problem has caused so much confusion.
Let’s Make a Deal: You are the contestant on a game show. You have to choose between three doors. One of the doors has the grand prize of an Automobile, while the other two have Goats behind them. Suppose you choose Door 1, at random. After you have chosen, but before you know whether you have chosen correctly, the host offers you a choice. The host opens door 3, and reveals a goat. He now gives you and OPTION to switch. The question is: SHOULD You SWITCH?
The standard analysis of this problem comes to the conclusion that you should switch. There are two cases where you start out with the goat. In both of these cases you will win the Auto by switching. There is one case where you start out with the Auto. In this case you will lose by switching. So, you have 2/3 chance of winning an Auto by switching, but only a 1/3 chance of winning if you stick with your original choice. We now proceed to explain why this analysis is wrong.
Our temporal probability model tells us that, BEFORE making the choice, you had a probability 1/3 of selecting the right door. After the choice has been made, probability has been extinguished. You either chose the auto or you did not. Furthermore, the Host KNOWS whether or not you have made correct choice. The game should be analyzed as a DETERMINISTIC game of STRATEGY, with asymmetric information. PROBABILITY is NOT INVOLVED in analysis of this game (as we will show).
Why Standard Analysis is wrong: The standard analysis assumes that host follows mechanical rule — he will ALWAYS reveal a goat, and ALWAYS offer a choice. This is nowhere stated in the description of the game. The host does not say anything about whether or not he always shows all contestants a door with a goat behind it. We will now develop the correct strategic game tree under the assumption that the host has two possible strategies. He can either open the door you have chosen to reveal whether or not you have won the prize, OR he can open an alternative door and offer you the option to switch.
Switching Leads to Loss: Analysis of this strategic game is easy. Whenever you choose a goat, the host should open the door and show you the goat. If you choose the automobile, it is a clever strategy on the part of the host to show you a goat, and to try to persuade you to switch. If you believe the standard analysis that switching is better, you will end up losing the auto.
Reality Check: This analysis is aligned with the portrayal of a game show in the movie “Slumdog Millionaire”. The contestant does not know the answer to the crucial Million dollar final question on the quiz show. The Gameshow host attempts to build trust, in order to deceive hero by giving him the wrong hint about the answer. Commenting on the probability analysis, the original host of the game, Monty Hall, demonstrated his mastery of contestant psychology. He won 8 out of 8 games, by showing the contestant the goat when he chose it, and psyching him into switching when the contestant made the right choice (NY Times: Behind Monty Hall’s Doors Puzzle). After the choice is made, probabilities are extinguished, and the game should be analyzed as a deterministic game of strategy, with asymmetric information.
Pervasive Confusions: Probability experts opine that confusions arise because “Our brains are just not wired to do probability problems very well.” This opinion has emerged from studies of human behavior in situations of uncertainty. In fact, the problem lies in the theories of rational behavior which are used to judge human behavior. These theories, which blindsided economists and investors to the global financial crisis are wrong. Failure of human behavior to match these theories testifies to the wrongness of the theories. Human behavior has evolved to judge probabilities and take actions which lead to highest survival probabilities. One of the key confusions created by probability theories which do not take the temporal nature of probability into account is discussed below.
Confusing Confidence with Probability: Before Occurrence of random event, probabilities are part of external reality. For a person who is IGNORANT about outcome, his KNOWLEDGE about the real world LOOKS like probability. It is just after the event, rather than before the event. Call this state of knowledge “confidence” to highlight its subjective nature. Despite similarity in appearances, confidence is NOT probability. You can BET on probability because external randomness should be same for all. You cannot BET on confidence because event has occurred, others may KNOW it. The enormous amount of confusion which surrounds the interpretation of confidence intervals exists because of failure to differentiate between pre-event probability and post-event confidence.
Concluding Remarks: In a study by Granberg and Brown (1995), cited in Wikipedia entry on Monty Hall Problem, only 13% of contestants switched, in actual experiments. Intuitive human behavior is smarter than Marilyn Vos Savant, reportedly the smartest human being (highest IQ) on the planet. The recommendation to switch is based on a probability model being applied to a situation where probabilities do not exist. This kind of confusion is common because standard probability theory models are NOT situated in time. Time flows in ONE direction. In particular, Bayes rule infers P(B|A) from the reverse probability P(A|B). This cannot be valid for events situated in time, since time flows in one direction only. It is valid to condition later events on outcomes of previous events, but one cannot infer probabilities of earlier events conditional on outcomes of future events.
Postscript: A valid criticism that can be made of the analysis above is that I am analyzing the wrong game. The original game is played according to the (admittedly artificial) rules that the host must show a goat, and must offer the contestant a choice. This criticism can be analyzed as a case of the confusion between confidence and probability. Let us suppose that the host plays mechanically according to the prescribed rules (this is not actually part of the problem statement in most formulations). Then the probability analysis is valid BEFORE the contestant makes a choice. AFTER the choice is made, probabilities are extinguished. Now we are in world of information and communication. The host has perfect knowledge about the outcome, and chooses to provide us with a hint about this outcome. Before this hint, we have CONFIDENCE (not probability) 1/3 that we have made the right choice. How should we update our confidence, in light of information provided by the host? It seems clear that the intentions of the host are crucial to how we should interpret a signal from the host. The question is simply not a question about probabilities at the stage where the host is providing us with information. It is question about whether or not the contestant should trust the host. We can achieve greater clarity regarding this issue by consider an ancient parable about a deadlier choice among doors.
The Lady and the Tiger: In an ancient parable, the King devises a clever plan to eliminate a commoner who has overstepped class boundaries to fall in love with the princess. Before a huge audience, the king asks the suitor to choose between two doors, one of which contains a hungry tiger, while the other contains a lady. Depending on his choice, the suitor will get eaten, or will wed the lady. In either case, he will be removed as a contender for the hand of the princess. The princess signals the suitor to choose door A. What is the probability that there is a lady behind door A? The answer is that there is no probability involved here. Either there is a lady or there is a tiger. Coming to the confidence issue, the value of the signal provided by the princess involves weighing whether the jealousy of the princess is stronger than her love. Probabilities are not involved in the decision.