My 3rd grader studied probabilities at school. At home I decided to demonstrate how it works in practice.

So we throw a quarter 50 times. 39 was head, 11 was tail, and I miserably failed to demo half/half rule.

What would be the proper way to demonstrate how probability works for 3rd grader?

• You could use a balance scale, and a bunch of pennies. Then flip the pennies, and put all heads on one side, all tails on the other? – GeoMatt22 Sep 28 '16 at 23:37
• Comments: (1) Perhaps the best thing at this age is to simply play games? Games with dice etc...? Third grade math covers multiplication, division fractions? Once he/she knows that stuff, you may be able to connect with the math? (2) Something was wonky with the coin flipping because only 11 heads or fewer among 50 flips has less than a .004% chance of happening by chance. (3) Let $x$ be the number of heads, $n$ be the number of trials, and $p$ probability of flipping heads. Law of large numbers is that $\lim_{n \rightarrow \infty} \frac{x}{n} = p$ it's not $\lim_{n \rightarrow \infty} x = pn$ – Matthew Gunn Sep 29 '16 at 0:24
• The "50:50 rule" is so often misinterpreted (e.g. as the gambler's fallacy) I wonder whether it might be worth avoiding. Perhaps it's more important to get a grip on the idea that more likely events tend to happen more often - perhaps home-made dice, with faces reading 1, 2, 3, 6, 6, 6, might make that point? And if you wanted to introduce the fractional reasoning, you'd also see that 6 tends to turn up on about half of all rolls (once your sample size is reasonably large). – Silverfish Sep 29 '16 at 0:38
• @Silverfish All kinds of games have non-standard dice (eg. King of Tokyo, a game strangely fun for adults too :P) – Matthew Gunn Sep 29 '16 at 0:44
• I like the idea of dice. It is perhaps too advanced, but something like rolling a pair of dice can get at the idea of combinations, i.e. 2 = 1+1 but 4 = 2+2 or 3+1 or 1+3. – GeoMatt22 Sep 29 '16 at 0:46

The problem with small number of tosses is that you may be inadvertently promoting avoidable fallacies. See for instance this post by Glen_b, regarding the "law of averages."

To show the law of large numbers a plot in something as available of Excel does the trick.

Here is a quick demonstration in Excel, really easy to set up. You can start off with the coin, then say something along the lines that the process is too mechanical for human consumption, and get your computer ready.

On the first column (A) generate $$250$$ values with RANDBETWEEN(0,1). On the adjacent column, get the cumulative average with AVERAGE($A$1:). Select the values in column B and generate a plot like this:

Make sure to emphasize that each toss is independent and completely unconnected to the prior, and to point out how far away from $$0.5$$ the results can be at the beginning of the experiment.

I remember the days, changing the plot, and starting with a small number of draws where the fluctuation in the plot is great. Have fun!

• Yeah, law of large numbers is that $\lim_{n \rightarrow \infty} \frac{x}{n} = p$ it's not the bizarre $\lim_{n \rightarrow \infty} x = pn$ (what many somehow seem to think?). The difference between the number of heads and the number of tails is essentially a random walk... it wanders off to plus or minus infinity as the number of flips goes to infinity.The law of large numbers may actually be a tricky concept for a third grader... it's actually a tricky concept for many educated adults! – Matthew Gunn Sep 29 '16 at 0:35
• Makes sense. First do no harm! And that's quite cool that you and your kids had fun with the simulation! – Matthew Gunn Sep 29 '16 at 0:41
• @MatthewGunn Your definitions (so concise and so critical) remind me of one of my very favorite posts on CV. – Blank Sep 29 '16 at 0:51