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I'm teaching an introductory statistics class at a community college, and we've just finished a unit on discrete probability. At the moment, the students' conception of the probability of an event A is

$$P(A)=\frac{\text{# of outcomes in A}}{\text{# of total outcomes}}$$

(Supposing all outcomes of the random experiment are equally likely to occur). We can use this basic understanding of probability to derive probability mass functions for discrete random variables, and graph these functions.

How do I help the students jump from the discrete to the continuous? Beginning with the basic understanding of probability as a ratio (required to understand the above formula), how do I move to the notion that the probability for a range of continuous values can be represented as the area under a curve called the probability density function?

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  • $\begingroup$ A person could TRY to motivate some geometric probability like "if we drop a coin on a grid of square tiles, what's the chance it will land crossing one of the cracks." (ask where the center of the coin lands and which locations would make it so the coin is crossing a crack. Then find areas.) Of course, this naive approach is inherently problematic [see Bertrand's paradox], and it's not clear if thinking in terms of area ratios will help understanding continuous distributions. But it's something I do for 20 minutes to motivate the need for calculus ideas when I teach calc-based prob. $\endgroup$ – Pat Devlin Mar 14 '17 at 15:03
  • $\begingroup$ Not long enough for an answer, perhaps I'll write one later: Use the relationship between the binomial distribution and the normal distribution. As the number $n$ of trials increases, a binomial distribution approaches a normal distribution with mean $np$ and standard deviation $\sqrt{np(1-p)}$. $\endgroup$ – Andrew Mar 29 '17 at 15:32
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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 picking up the idea by just flat-out being explicitly told ("the area under a curve is the probability of getting an outcome in that range") and then practicing some with a normal-curve table. Start with a very simple case: normal curve with mean 9, standard deviation 2 (sketch); what is the probability of getting a value above 9? Every class I've taught, at least one student has intuited that the answer is 50%; and I can say, yes, because half of the area is above 9. Run more exercises and in every case sketch the curve and check for reasonability.

I teach out of Weiss Introductory Statistics and it works perfectly fine. Up through the 8th Edition there wasn't any text on continuous probability distributions in the general sense at all; it just started using a normal curve table like this in Ch. 6. As of the 9th edition they added a 2-page introduction on density curves in general, but I don't find that any more illuminating; I just skip it. In fact -- I even skip the whole chapter on discrete probability distributions (Weiss marks it as optional). Due to the conceptual gap here between discrete/continuous distributions, I don't think the discrete case increases understanding of the continuous case, and it delays getting to the crown jewels of the course: inferences with confidence intervals and P-values for means and proportions for large samples.

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This is treason, but anyway:

If your students can jump from "ratio of outcomes in $A$ over all possible outcomes" to "ratio of length of interval, over total feasible length", then the answer why probability can be represented as the area under a curve could half-jokingly (but only half-) be "for convenience, since we set the height of the curve at a value that it will replicate our "ratio" approach".

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The "ratio approach" would require that

$$P([a,b]) = \frac {length[a,b]}{length[a,c]} = (b-a)\cdot \frac {1}{c-a}$$

Instead of doing a division in one dimension, we do a multiplication in two dimensions, by setting the probability density curve at height $1/(c-a)$.

With any other distribution, one has to move from simple multiplication of the sizes of a rectangle, to integration to find the area under a curve.

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For discrete distributions, it's helpful to look at coinflips. You can find the probability of getting m heads/tails out of n flips. (Though a normal distribution may be simpler to start with than a binomial distribution.)

As for continuous distributions, it may be helpful to look at the Maxwell-Boltzmann Distribution, which can tell you the probability of finding a particle traveling at velocity v (where v is the random continuous variable). Since the total probability is one (100%), it means that the area under the curve described by the function is also one, which can be exploited to normalize the function.

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    $\begingroup$ Thanks for pointing that out, just edited. $\endgroup$ – MPath May 19 '17 at 19:10
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Draw a square on the board and shade in half of it. Ask your students, "If you throw a dart and hit the board, what's the probability that you hit the shaded region?" Cut the region down to a quarter of the square and ask the question again. Lead this into a discussion of how the probability is the ratio of the area of the region to the total area and tie this back to the discrete probability formula.

Now ask the same question about a point. A point has "0 area" so the probability of hitting it is 0. When there are infinitely many possibilities, i.e. there are infinitely many points inside the square versus, for example, a finite number of die rolls, we can't talk about the probability of individual events - just ranges of events/results.

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I present just a couple of ideas that might make a bridge between the discrete and continuous.

Firstly, one thing I notice with many students is they don't have an idea that the probabilities of all the outcomes add up to 1. This stops them being able to do a lot of calculations for continuous distributions later. Moreover, it means they don't get the fundamental idea of distribution which is a description of all the possibilities and how likely they all are.

I think that we should probably never ask for the probability of any event in isolation but always also ask for probabilities for a set of events that complete it to the whole sample space. For example, don't just ask for the probability of getting a 7 on two dice, but also less than 7 and more than 7. At the very least, always ask for the probability of A and not A. My instinct is that to start with, students will calculate both based on the ratios, and maybe later will realise there's an easier way. But at least we'll drive home the idea that events never exist without other events to fill them out.

Secondly, perhaps a road to asking about continuous probabilities, is to ask about a discrete set of objects with continuous measurements on them. If the measurements are written with decimals it will be even more obvious that they are supposed to be continuous.

For example, take 14 trees with their heights measured in feet:
60.92 63.39 64.10 59.07 63.05 59.64 60.07 60.69 60.28 61.62 58.49 56.81 56.43 59.49 (from the Loblolly dataset in R).
Then you could ask questions like the probabilities of choosing a tree with height in the ranges 55 to 57.5, 57.5 to 60, 60 to 62.5 and 62.5 to 65 including drawing a graph showing these probabilities. Asking various different collections of zones might help to necessitate the axis being marked out on a scale rather than in discrete labels. Choosing zones that aren't equal might make it easier to necessitate a way to show the amount of probability in a way other than height so that the graph isn't so misleading. A dataset with 100 numbers in it might make this even more obvious and lead towards the probability distribution which has infinitely many numbers.

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