50

For some reason, the 'extend it to 100 doors and eliminate 98' explanation doesn't make it any clearer for me. Rather than talk about probabilities as fractions, I explain it this way: "If you picked the car (without knowing it) on the first choice, you'll lose it by switching, whereas if you didn't pick the car, you'll gain it by switching." (stop here ...


25

Your explanation, by the way, is very elegant. As an experienced mathematician, I see immediately that it cuts right to the heart of the matter and admits no ambiguity. Unfortunately, this is precisely the quality that makes it unconvincing to others; the main confounding aspect of Monty Hall is that it ruthlessly exploits an intuitive misunderstanding of ...


24

Firstly, don't forget that your student has thought hard to come up with his answer and to be told it is wrong may be taken as invalidating his effort, or even insulting his intelligence. This might be at least a small part of his resistance to accepting your response. I would have recommended starting by asking him to explain more about his thinking, and ...


18

There isn't any sure-fire method of explaining anything, and especially in math. But specifically in the case of the Monty Hall problem it has been proven by extensive experience that many individuals with otherwise above average intellectual capacities exhibit an exceptional tenacity in refusing to accept the (otherwise) widely agreed upon solution; don't ...


15

Perhaps it's not the explanation that's the problem. I suggest you have them explain to you their understanding of the problem. Listening to their justification might reveal why your explanation is not gaining traction. Even in the cases where people are saying they now agree with you, you don't necessarily know that they understand the problem. It's ...


14

I've had best luck by simplifying the problem to asking whether you want to pick one door or two doors. Everyone understands they'd be better off picking two doors. At that point I tell them to just ignore the fact that Monty showed them what was behind one of the doors they didn't pick. They're still picking either one door or two doors. Edit: The more ...


14

The answers provided here so far give lots of good tips but I think they're not addressing a key part of the question, which is "why do we need to count two events (50,52) and (52,50), instead of one event (50,52)?" The answer is that you can do it either way. The textbook way is counting $$P(X=50 \cap Y=52) + P(X=52 \cap Y=50) + P(X=51 \cap Y=51) = P(X=...


11

I found the most helpful way to think about the problem is to expand it to a larger number of doors. For example, if you have them select from 100 doors, where 99 are losers and 1 is a winner. Then after the initial selection is made, eliminate 98 doors. The crux of this explanation is from one of Polya's heuristics, as explained below. The initial answer ...


11

As you told the student, the easiest way is to regard the Lebesgue integral as beginning with a partition of the range, rather than the domain. Perhaps a more refined way to view this is that the partition, rather than the "heights" of the rectangles, can be used to encode the "shape" of function being integrated. The way to encode the function in the ...


10

One often sees the phrase "a fair coin." So, if you wish, this is a theoretical object which does not exist in the real world but the name is evocative and useful. In real life a (real) coin flip is used to decide some things (football). One may object to an element of chance in a contest of skill. Perhaps the side winning the toss gets an unfair advantage. ...


9

I think there are a few independent questions here that can be considered teaching mathematics (I will not broach the subject of how to solve the problem). First, there is: How can you state this problem unambiguously without using artificial language? Indeed, as I see it, the reason this problem sounds paradoxical is that the phrase "one of them is a ...


9

Ask whether they think putting the marbles in a bag into any particular arrangement would affect the outcome. If they're ok with this, have them consider the arrangement of Bag 2 where the marbles are arranged in 10 wbb groups. They're likely to agree that the hand now has 10 equivalent groups to pick from, and that each group yields the same odds of getting ...


8

I never understood it until my Maths lecturer explained it to me. Unfortunately I can't remember the exact way he explained it, but I'll try my best to remember. I really like your own explanation, so there is a chance that the following won't work. In that case, you may have to resort to empirical experiment. Another observation from my own experience is ...


8

The best sure-fire method for teaching this to people who don't want to learn it is to set up a Monty Hall style game for small stakes of real money. When they start losing 2/3 of the time, they will become more receptive to your explanation.


8

DavidButlerUofA's answer hits the spot on pedagogical methodology. However, the actual question is about establishing the correct intuition. With that said, a good way to falsify the wrong intuition is to enlarge the bag to contain millions of marbles; or rather, to shrink the marbles until they are black and white grains of dust. With 2/3 of the dust ...


8

If you "stay" then you win when the prize is behind the one door your originally selected, yet when you "switch" you win when the prize is behind one of the two doors you originally did not select.


8

I'm not sure if you consider the board game Monopoly as a real-world example, but it is often used to explain Markov Chains to laypeople. Ian Stewart has a couple of Mathematical Recreations articles about it in Scientific American. The first (in the April 1996 issue) has the title "How Fair is Monopoly?" (A copy it can be found here.) The second (in the ...


7

I'm going to suggest what you're considering is very difficult. And also, I don't think it's something I'd want to do. (But we'll do it below anyway, for fun!) Fractions are an early concept that involves a multiplicative relationship between things. In the case of fractions, one of those things is a whole. Students may have some understandings of fraction ...


7

One suggestion that might be worth considering is asking the students the same question but for a bag with no white balls. Suppose you have a bag with 1 black and no white balls ( probability of picking black = 100% and difference between white and black = 1) Now suppose you have a bag with 10 black and no white balls ( probability of picking black = 100% ...


7

I wrote a long answer to this, which then made me realise that I think this needs the rule. [Restart:] A few pointers first: Don't number the bags if you have numbers of marbles: big/small, black/brown, Peter/Paul, A/B. Don't actually use bags when you want to see what is going on: bowls, or for class experiments transparent vases. Marbles are fine, since ...


7

Textbooks tend to try to use the language that would be familiar to readers. In natural speech the use of "likely" is much more prevalent than that of "probable". Interestingly, though, according to Google N-Grams, that hasn't always been the case, with "probable" and "likely" starting nearly level in popularity in 1800: And yes, as you mentioned, as ...


7

I see at least one good topic for this: Monte-Carlo algorithm to compute area or volume. It has a strong geometric side, is definitely probabilistic, only involves simple concepts, and is even relevant to real-life (of mathematicians). Imagine we want to compute the area of a region in the plane, say a oddly-shaped pool in a fancy California hotel. Problem ...


7

This is an uncomfortable moment, mathematically, in a non-calculus-based statistics course; frankly, we simply need to steal the calculus concept and hope that students trust us about it, without formal grounding. It's somewhat degenerate mathematics but it's the position we're required to deal with. That said: I find that students do a pretty good job of ...


7

This is a very good question. The issue comes up frequently. I explain this using a toy model: throw two regular six-sided die. What is the probability that the sum is 3? With some physical modeling, you can become convinced that the answer is 2/36=5.5%, This corresponds to the two possibilities 1+2 and 2+1. But why are these two possibilities distinct? We ...


6

Speaking from my own experience, a stage I had to go through before understanding any explanations of the logic and math involved was indignation at being baffled by a cheap fairground trick that probably goes back centuries. Also the eventual recognition that what put me, the punter, at a disadvantage to the operator was the fact that s/he is working to a ...


6

While your explanation is correct, and easily understood by people like us, it's a bit too terse (a quality we like) for a lay person to understand. You can simply expand on it. And most importantly ask the audience questions along the way. Every explanatory sentence should have a question that goes a long with it. The key is for you to identify where the ...


6

I find that most people who THINK they understand the Monty Hall Problem, actually don't. For about 5 years I was one of them, until a further insight made me understand it better. More of that at the end. However, I came up with a super-exaggerated version that seems to give people pause for thought at least. In the UK, our lottery has a probability of 14,...


6

Here's my take on explaining it. (And it usually involves a drawing for me :-) ) Assuming the host opens a door the odds of making the right choice without switching the door is 1/3 because it was made before having the knowledge the host provided. Once you decide to make the switch here is a breakdown of the probability of losing VS winning:


6

I definitely think you can get the intuitions behind probability to elementary age kids, and doing so might even be a way to introduce the important notions of equivalent fractions / related rates / etc. Take a six-sided die and mark 1 on one side, 2 on two sides, 3 on three sides. Show the kids the die. Ask them, if you roll this die a bunch of times, ...


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