Logic v the crowd!

In September 1990, Marilyn vos Savant, known for her “Ask Marilyn” column in Parade magazine and her record-high IQ, received an intriguing question from reader Craig F. Whitaker about the now-famous Monty Hall problem.

The problem

The question was something like:

Suppose you’re on a game show, and you’re given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what’s behind the doors, opens another door, say No. 3, which has a goat. He then says to you, “Do you want to pick door No. 2?” Is it to your advantage to switch your choice?

The answer

Vos Savant confidently answered that contestants should switch doors, explaining this would increase their chances of winning from 1/3 to 2/3. Her response ignited a firestorm of controversy, leading to an unprecedented flood of approximately 10,000 letters to her column. While not all were objections, a significant portion vehemently disagreed with her solution. The sheer volume of mail was staggering, highlighting how deeply the problem resonated with readers.

Among the critics were professional mathematicians and academics. One PhD mathematician dramatically declared, “You blew it, and you blew it big!” while another condescendingly suggested she consult a probability textbook. Most argued that after the host opens a door, the probability becomes a simple 50/50 choice between the two remaining doors.

Undeterred by the criticism, vos Savant stood her ground. She devoted three subsequent columns to explaining her solution, using various analogies and explanations to clarify the counterintuitive nature of the problem. Her persistence in the face of widespread disagreement was remarkable.

As the debate raged on, it gradually became clear that vos Savant was indeed correct. Through continued explanation and even practical demonstrations, many critics, including some of the mathematicians, came to understand and accept the correct solution. The controversy had brought the Monty Hall problem into the public spotlight, transforming it from an obscure probability puzzle into a topic of widespread discussion and debate.

This episode served as a powerful reminder of how even seemingly simple probability problems can confound our intuition. It highlighted the value of rigorous mathematical thinking and the potential pitfalls of relying too heavily on gut feelings when it comes to probability. The narrative of vos Savant and the Monty Hall problem became a classic example in discussions about probability, human reasoning, and the occasionally humbling nature of mathematical truth.

The mass response, particularly the large number of objections from both laypeople and professionals, is what made this incident so notable in the history of the Monty Hall problem. It demonstrated how deeply counterintuitive the correct solution was, even to many people with advanced mathematical training. This controversy ultimately contributed to making the Monty Hall problem one of the most famous and discussed probability puzzles in popular culture.

What it means?

The Monty Hall problem and the controversy surrounding Marilyn vos Savant’s answer served as a powerful demonstration of mathematical logic.

This incident illustrates several important points:

1. The power of mathematical reasoning: Despite widespread disagreement, even from those with mathematical backgrounds, the correct solution stood firm because it was based on sound probabilistic logic.
2. The fallibility of intuition: Many people, including experts, initially trusted their intuitive understanding of the problem, which turned out to be incorrect. This shows how our instincts can sometimes lead us astray in probability scenarios.
3. The importance of rigorous analysis: The controversy highlighted the need for careful, step-by-step reasoning in solving probability problems, rather than relying on quick judgments.
4. The value of persistence in the face of criticism: Vos Savant’s steadfast defence of her correct answer, despite overwhelming opposition, ultimately led to greater understanding of the problem.
5. The potential for learning and growth: As the debate unfolded, many people, including some initial critics, came to understand and accept the correct solution, demonstrating the power of open-minded inquiry and the ability to change one’s views in light of compelling evidence.
6. Crowd sentiment: The Monty Hall problem controversy serves as a compelling reminder that crowd sentiment, no matter how widespread or passionately held, cannot override mathematical truth.

Conclusion

The Monty Hall problem illustrates how mathematical truths can sometimes appear to contradict our everyday understanding of the world. This disconnect is often at the root of many misconceptions in probability and statistics.

When thousands, including respected mathematicians, insisted that Marilyn vos Savant was wrong, it wasn’t their collective conviction that ultimately mattered, but the immutable logic of probability theory. This incident demonstrates the unwavering nature of mathematical principles in the face of popular opinion. It underscores the importance of rigorous analysis over intuition or consensus, especially in fields like probability where our instincts often lead us astray.

The resolution of this controversy highlights a fundamental strength of mathematics: its ability to reveal objective truths that persist regardless of how many people initially disagree with them. In a world where popular opinion often holds sway, the Monty Hall problem stands as a testament to the power of mathematical reasoning to cut through misconceptions and reveal unexpected realities.

While common sense and intuition are valuable in many areas of life, they can be unreliable guides in the realm of probability and statistics. This case serves as a powerful reminder of the importance of grounding our reasoning in solid mathematical principles, especially when dealing with counterintuitive scenarios.

It is a testament to the power of mathematics that it can lead us to truths that our intuition might never reach on its own.