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I asked this question in a class of 13-year-old students: Bag 1 contains 2 black marbles and 1 white one. Bag 2 contains 20 black marbles and 10 white ones. Which bag is more likely to yield a black marble? Or maybe the chances are equal?

We were discussing this problem informally and intuitively (the probability as a ratio is not introduced yet). One of the students conjectured that when the ratios of number of blacks to the number whites are equal in two bags then the bag with greater difference between blacks and whites is more likely to yield a black one (a known misconception, as mentioned here).

How can I convince him that his answer is wrong? I simulated the situation with Mathematica for 100,000 draws, for many times. However it made him doubt his conjecture just a tiny bit!

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    $\begingroup$ Just a note on terminology and keeping everything clear. You say Bag 1 contains 2 black marbles and 1 white one. That can be a mouthful and a brainful..why are the bags numbered when the question is about numbers? Why not "small bag" and "large bag"? "Marbles" is a dated reference; what other things can't be distinguished just from feeling that you'd pull from a bag that 13-year-olds know? Just pointing out that with kids--or many adults--"you lost me at 'bag one has one white marble'" is totally possible. $\endgroup$ – HostileFork Nov 3 '14 at 20:46
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    $\begingroup$ "How can I convince him that his answer is wrong?" The better question might be "What mathematical concept do I want them to grasp?" I say "better" only in the sense that it will lead you toward your answer. Finding a way to convince students that their answer is wrong does not necessarily help them grasp a concept. So your first step is probably to identify what mathematical understanding you are getting at. $\endgroup$ – JPBurke Nov 3 '14 at 21:00
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    $\begingroup$ Is it a bad thing that a 13 year old is not convinced by 100,000 simulations whereas many 17 year olds would be convinced by 100 simulations? $\endgroup$ – BCLC Mar 16 '17 at 15:01

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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 withholding judgement for a little longer. Sounding like you are genuinely interested in learning more about his ideas will help (actually being interested will help more, of course ;) ). Another helpful thing to ask him is how he might go about checking if his intuition is right, rather than presenting him with your simulation first up.

The misconception might be caused by thinking about repeated trials without replacement. He may be thinking about taking a handful of marbles, or one after the other. In that case, with the second bag, it's possible to pull out many black marbles in a row, but in the first it's not. Also in the first bag, you only pull out two before you're guaranteed a white, whereas in the second you have to pull out 20 before being guaranteed a white. Even with formal probability, it actually is more likely to pull out two black in a row from the second bag! If this seems to be the issue, my suggestion is to ask him what would happen if you only had one chance to pull out one marble. Would it still be more likely in the second bag then?

A computer simulation may not be convincing to him, simply because it's a computer and therefore not real. It would be much more real if he did it with actual bags of actual marbles (or beads or m&ms or whatever). This is more likely to be the suggestion he comes up with himself if you ask how he could test his understanding. To really drive home the the "only one chance" idea, I would get everyone in the class to give it a go once. Even better, give everyone their own bag and all pull out one marble simultaneously. Then, if you do do the simulation later, you can say you're simulating ten thousand different people all having their own one and only chance.

The only other suggestion I have is to investigate it not just for 1:2 versus 10:20 but also for 2:4, 3:6, 4:8, 5:10...The weight of evidence from all of these will be more convincing, and highlight more strongly that it's just the ratio and not the difference that is the important thing.

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    $\begingroup$ +1 The point that it is more likely to get a black-black pair from the large bag is important. The original question implied taking just 1 marble but didn't give that aspect of the problem the same focus as other aspects. $\endgroup$ – Larry K Nov 4 '14 at 7:02
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    $\begingroup$ Also very good point on the pedagogy of how to correct the student. I remember the "ego sandwich" -- you need to help the ego before and after feeding the student the "meat" of the correction. $\endgroup$ – Larry K Nov 4 '14 at 7:20
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    $\begingroup$ This is excellent pedagogy. It's just as important to figure out what the student is thinking rather than just if they're right or wrong. Plus, as a former science teacher, I'm biased towards experiments in the classroom. $\endgroup$ – Mark H Nov 4 '14 at 8:09
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    $\begingroup$ +1. I run a similar experiment with my students with coins and it works far better than using computer simulations. $\endgroup$ – Pablo B. Nov 4 '14 at 8:14
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    $\begingroup$ Good answer, but before rushing into an experiment, a teacher should use the TCL to estimate how much experiments should be done to avoid random fluctuations to mess up with the point. Using more than two point, as advised, is all the more important (it is unlikely that random fluctuations will be monotonic - but how unlikely exactly?). This turns out to be a good example of how much more teachers have to know than what they teach. $\endgroup$ – Benoît Kloeckner May 19 '17 at 10:04
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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 a black marble as bag 1.

This is a really neat problem, by the way.

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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 grains being black, I bet the student will sense that the probability of extracting one black grain is 2/3. "Clearly" the numerical difference (which is in the millions) is not relevant.

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    $\begingroup$ This is interesting. Somehow we have different neural architecture for discrete and continuous probability, and you are saying "when the discrete intuition fails, bind it to the (probably correct) continuous intuition". I like this a lot. I use a similar idea to argue that the exterior angles of any convex polygon sum to 360: extend the exterior rays, and "zoom out", so the polygon is basically just a point. Then you just have a bunch of rays emanating from a point, and the sum of the angles must be 360. $\endgroup$ – Steven Gubkin Nov 6 '14 at 14:52
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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% and difference between white and black = 10)

The second case obviously has a greater difference as your student suggested might be a determining factor but clearly the probability of picking a black ball in this scenario is the same as when the difference was = 1.

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    $\begingroup$ This appeals to me as a mathematician, but students often have trouble accepting "extreme" circumstances. This is a good way to stretch their minds to accept such arguments, so +1. I would pair such an argument with a more "reasonable" explanation, such as NiloCK's. $\endgroup$ – Steven Gubkin Nov 3 '14 at 16:23
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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 they are familiar yet insignificant to them, but in most sets they're not black and white. (For the class, use ping pong balls in e.g. orange and white.)

In private discussion (Class discussion is more about what the right answer should be.), I guess he'd bring the marbles and you the bowls and together you study it from the very small scale up.

But, and this is what I referred to above, I think this is going to need a systematic approach, and that means deriving that you count all balls, and count the part of them that you're after - probability ratio. You can make it special with "I haven't told this in class yet ...", but I think that to effectively scale between 1 and 2, 2 and 4, and 10 and 20 and make it stick, you'll have to go that way.

For the completely different alternative you have the runway coloured in black sections and white sections and the pilot of the biplane losing his shawl (etc.) somewhere over them. Just as random as a draw, but without the confusion of lack of order. (Coloured fence, horses on the merry-go-round, any random selection of a sectioned surface would work.)

And don't worry. You're a good teacher, as you know when to ask for help. So be assured that you will get the subject across to him.

                                                                                                                                      

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Describe hitting bag 1 with a hammer three times.  Stipulate that each blow shatters one of the three marbles into ten equal-sized pieces.  Ask them whether they think the probability is different now.

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As no one has mentioned this yet, I should note that the cumbersome question "Which bag is more likely to yield a black marble?" is yet another linguistic convolution that is more than likely to yield an unnecessary distraction for the 13-year-old adolescent who is simply trying to grasp the crux of the matter.

No 13-year-old uses the world 'yield', as might be the wont of many an ancient mathematician. Use plain language.

For example:

One bag contains 3 balls - 2 black and 1 white.

A second bag contains 30 balls - 20 black and 10 white.

You are blindfolded and pull one ball out of each bag.

From which bag are you more likely to get a black ball?

And, by the way, repeated simulations on Mathematica is to any kid just another meaningless black box that won't convince him of anything. Indeed probably not just to kids. Mathematicians who couldn't believe the Monty Hall solution would not have been convinced by Mathematica simulations either.

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The 13-year old is "right" -it is the question that is wrong. It is a classic example of an attempt to create "tangible situations" in order to present abstract concepts, forgetting in the process that "tangible situations" introduce aspects that have been abstracted out in order for the abstract concept to work, but only by those that are already familiar with the abstract concept.

But the 13-year old is not "already familiar" with the concept, and so it uses plain logic, still unaffected by what we are now trying to teach him.

So it does not think of abstract numerical ratios $2/1$ and $20/10$, or $2$-in-$3$ and $20$-in-$30$. Instead, it pictures one bag with 3 marbles, and another bag with 30 marbles,with colors distributed as mentioned. Probably the last thing it will pay attention to is the relation between the ratios or the proportions. Other things may arise in its thinking: for example, the size of the bags. Are they of equal volume? If yes the one will be much more empty than the other... will this affect its chances if it puts its hand in? Is anybody telling the 13-year-old that this aspect of the "tangible situation" won't affect its chances? And if this is the case, why?

Are the bags such that they really allow for homogeneous layers of marbles? If the bag is made of soft leather, the one filled with the 50 marbles will tend to create an asymmetric heap, with larger base than tip... does this affect the actual real-world chances of drawing a black marble from the bag? This aspect is also intended to be "abstracted out", but did anyone tell that to the 13-year old?

Because its logic will likely try to assess all these aspects of the "tangible situation" in order to answer the question, an then... all bets are off, as regards what it will appear as the correct answer to its mind.

In short, when creating such examples, we must pay great care so that they are as "neutral" as possible, leaving the information we want the answer to be based upon (here the ratios/proportions) as the main piece of information coming out of the example, so that we increase the chances that the student will indeed focus on them, and not on all these other unintended aspects our example may introduce.

I am not saying this is easy, on the contrary it is hard to very-hard, because we have to temporarily put aside our own familiarity with the concepts we are trying to teach.

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13 year olds? Marbles? The answer is easy (as was suggested by David Butler) -- do the physical experimental test of the probabilities.

And take the opportunity to expand your teaching beyond the symbols of mathematics and into physical experimentation: teach the students how to keep a lab journal with the experiment, experimenter, results, additional factors, etc all recorded, signed, and dated.

As someone with an engineering degree, I appreciate both the pure value of mathematics and its application to the physical world. I humbly suggest that your classes would become even more interesting if you seize opportunities to apply math to the physical world whenever the opportunities present themselves.

A possible source of professional collaboration would be your friendly local physics teacher. She or he might have some fun physical experiments that you can then use to explore the mathematics of the experiment. (Or do the math first and then the experiment.)

You could also discuss how the great mathematicians / scientists of the past would work on both the symbolic and experimental--get your history teachers involved too!

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How many marbles are you going to draw?

My initial answer was like the student's answer. Since you implied by the question that there was a difference, my thought pattern went like this:

If you draw one marble the odds are equal. (1 in 3)

If you draw two marbles (without replacement), the small bag ahs a 100% chance of yielding a black marble.

To convince the student I would start by acknowledging the question is a bit ambiguous (it never hurts to give his ego a shot). ... Then try the question over with a bag with the same number of marbles of each color (say 5 of each). Ask the student "If I draw one marble from this bag, is it more likely to be black or white?" When he says there is equal chance, double the number of marbles in the bag but keep them 50-50 and ask again.

If he sees that the proportion is the important thing here, he might see it for the other case as well.

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I think you tell him to write down and remember his answer to this question, then you move on to teaching probability and how to calculate it. Some time later when probability is understood ask him to calculate the probability of drawing a black ball from each bag, then see if he wants to change his answer, hopefully he will.

This will give him the satisfaction of knowing that he has learnt and understood something useful that was not clear to him before. It also allows him to correct his own misconception which is more positive than trying to explain to him why he is wrong.

After that you can ask them if he draws two balls which bag is more likely to yield one white and one black ball.

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    $\begingroup$ Knowing how to calculate probability in textbook problems is not the same as understanding probability. How do you connect the abstract concept of probability with the intuitive understanding of chance? $\endgroup$ – Dag Oskar Madsen Nov 4 '14 at 0:19
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    $\begingroup$ The question was about how to address students' misconceptions. $\endgroup$ – Dag Oskar Madsen Nov 4 '14 at 9:55
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Nice question! Two ideas:

(1) Compare A: {2 black, 1 white, difference = 1}, to B: {100 black, 98 white, difference = 2}. Should B have a greater chance than A of drawing black?

(2) Compare A: {2 black, 1 white, difference = 1}, to C: {1 black, 0 white, difference = 1}. Should A and C have the same chance of drawing black?

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    $\begingroup$ This could be a good start but might not convice them fully (if I understood the question correctly) as they only think the difference is relevant "when the ratios of number of blacks to the number whites are equal in two bags." $\endgroup$ – quid Nov 3 '14 at 13:40
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    $\begingroup$ @quid: I missed the equal-ratios stipulation; thanks. So I only addressed why the B-W difference is not the key. $\endgroup$ – Joseph O'Rourke Nov 3 '14 at 14:32
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This answer is assuming that the question is talking about the probability of drawing EXACTLY 1 marble out of the bag and determining the likelihood of it being a black marble or being white marble.

Now, something that many of the school teachers who teach math, (no offense to anyone) including the one who "taught" me probability is that they always leave out the idea that math is a formal system that is always based on fundamental assumptions which mathematicians call Axioms.

First off though, let's start with defining the word probability: one of the dictionary definitions is "Probability is the extent to which an event is likely to occur, measured by the ratio of favored cases to the total number of possible cases. Or using mathematical notation, given event E:

$P(E)=\frac{favorable\:outcomes}{total\:possible\:outcomes}$

The idea behind this is that we are making the assumption that each individual outcome is equally likely. That means that we are making the assumption that there exists a probability for picking a single white marble such that the probability of doing so is exactly equal to the probability of picking a single black marble. Or, expressed in mathematical notation, P be predicate for "Probability of picking", let w be predicate for "white marble", let b be predicate for "black marble": $\exists \:P\left(w\right):\:P\left(w\right)=P\left(b\right)$

Also, we represent the probability of an event that will NEVER happen, as a 0% chance of happening meaning that for every 100 times the event could happen theoretically, it happens 0 times out of 100 and the way we represent this is:

$P\left(E\right)=\frac{0}{100}=0$

And, in a similar fashion, if the probability of an event is CERTAIN, then it will happen 100 times for every 100 times it can happen i.e.

$P\left(E\right)=\frac{100}{100}=1$

If we use the definition of certainty being 100%, i.e. 1, and we assume out of all the possible outcomes, at least one of the individual outcomes must occur (in this case a person has to either draw a white marble or black marble, can't draw zero marbles or have a random grey marble magically pop into existence), Then the sum of all the individual outcomes must equal to 1.

If you let A,B,C ... be the individual outcomes then mathematically: $P\left(A\right)+P\left(B\right)+P\left(C\right)+...\:=1$

Thus if you combine the assumption that each of the individual outcomes are equally likely and the definition that all individual outcomes must sum up to 1, then if you number all the white marbles (from 1-10) and all the black marbles from (1-20) then you get:

$P\left(w_1\right)=P\left(w_2\right)=...=P\left(w_{10}\right)= P\left(b_1\right)=P\left(b_2\right)=...\:P\left(b_{20}\right)$

and you also get:

$P\left(w_1\right)+P\left(w_2\right)+...+P\left(w_{10}\right)+P\left(b_1\right)+P\left(b_2\right)+...\:P\left(b_{20}\right)=1$

Solving this gives you:

$P\left(b_1\right)=\frac{1}{30}$

and since there are 20 black marbles, if you let E rep. the event of selecting a black marble you get:

$P(E) = \frac{20}{30} = \frac{2}{3}$

and, you would use the same assumptions and reasoning for the other bag with just 2 black marbles and 1 white marble to get the same probability.

Now, the KEY IDEA here is that we made the assumption that each individual outcome is equally likely. This is not necessarily true in real life. For example, if you toss a coin (and actually depending on how you toss the coin) the probability of heads is actually slightly higher than the probability of tails (51% and also there's a very small probability of landing exactly in the middle). Another example would be if you had a loaded die, then the probability of each number wouldn't necessarily be $\frac{1}{6}$.

The simulations that run in theory will not necessarily have the same results as actually conducting the experiment in practice because real life is usually not as ideal and beautiful as it is in mathematics.

In my opinion, I believe the best answer to the question "which bag is more likely to yield a black marble?" is: "It depends on how you draw the marble out of the bag, depends on the composition of each marble (maybe like some marbles are heavier than others) etc. But, if you assume the likelihood of drawing each individual marble is equally likely then, you should have an equal chance of drawing a black marble from bag 1 as you do in bag 2."

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Well, why does he think his answer is right?

'Don't criticize what you can't understand.'

If he gives an answer involving fractions, then perhaps there is a simplification error (Unlikely to have a simplification error so it's unlikely the answer will involve fractions)

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