59

Your child is using the Principle of Insufficient Reason, which states that if we have no information about something other than the set of possible outcomes, then we should assume that all outcomes are equally likely. This principle is behind basically all of statistics, probability theory, and statistical mechanics, although it is often disguised in some ...


52

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 ...


43

I don't think that talking about probabilities formally would be to any benefit for your son. However, you could simulate a lottery at home, using a die. Say that a player wins if they guess right the next outcome. So, at each roll, there is a $1/6$ probability that the player wins. Roll the dice many times - say 50 or 60 - and write down the number of wins. ...


26

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 ...


22

I think so far best reaction is the top-voted comment: Have you asked him to explain what he thinks “probability” means? I'd address the topic from here. And as this is not a school environment when teacher has the truth and shares it with the student, I'd take the approach to try to understand him through questions: What probability means for you? Are ...


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 ...


16

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=...


12

Without going into the mathematics to deeply I would say it boils down to this: There is only one way of winnning the lottery: Guessing all the numbers correctly. But there are a lot more possible ways to lose. So his "whole premise of his reasoning is wrong" as you say because he is ignoring millions of possible outcomes!


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. ...


10

A slightly different approach: Let's say there are 100 lottery tickets in total and there is only one ticket that will win you the prize. If you don't buy any tickets at all, what's your chance of winning? No chance. Nothing out of a hundred is zero %. What if you could buy all the tickets? You are certain to win. You've got 100 out of 100 tickets. Your ...


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 ...


9

Taking the ed part of the question: Don't feel like you have to convince the kid of everything immediately. Give him time. In particular, watch out for him just trolling you. If you do decide to engage him, you could do so by suggesting that if he really thinks it's 50-50, that you'll take the opposite side of the bet for 2:1 odds. Ask him to put some ...


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 ...


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 ...


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