# Law of large numbers as a middle school topic?

My daughter (a biologist) is presently teaching also math at a middle school (9th grade, so about 14 years olds). Now the topic in probability seems to be the law of large numbers! More and more I doubt that is a good topic for that age group.

• It is a limit result, the children have no prior experience with limits or asymptotes ...
• how to make good relevant exercises?

I was shown an exercise meant for the children: First, they are asked (from a drawing of all outcomes) to find the probability of two head when throwing a fair coin 4 times. Then they are asked what they believe the probability of exactly 500 head when throwing 1000 times (about 2.5%) ... I find this a strange way to do it! So, two parts to the question:

• If you must introduce the law of large numbers in 9th grade, how?
• Is that really appropriate?
• Most math can be taught at a "what it means and how to use it" level without rigorous formalisms. You don't need a formal definition of a limit -- or even an informal one -- to understand the law of large numbers. Wikipedia's article opens with a graph that demonstrates the law quite nicely, and not an epsilon or delta in sight.
– Mark
Feb 15, 2023 at 3:52
• Agreed that's a bad exercise if it's intended to highlight the law of large numbers. Feb 15, 2023 at 4:41
• @Mark I'm not sure that graph shows what it means or how to use it. I don't think LLN can be meaningfully understood without a limiting process. A statement like "if you do enough trials you'll approximate the true mean" literally cannot be used in the sense of LLN - because there is no information about the rate or quality of the approximation. Something like Chebyshev's inequality (or other quantifiable approximations) would be much more useful to someone without knowledge of a limit process. Feb 15, 2023 at 10:55
• @Mark: One problem I see with using graphs is, at this grade level (at least in Norway) the pupils are not really used to graphs, they need to see graphs with some more familiar numbers ... Feb 20, 2023 at 0:48

You have two different questions.

1. Should?

I don't think your daughter has a choice here. So it's irrelevant to her. Obviously society has a choice. I don't think the topic is so awful, personally. It can be understood intuitively without formal limit training. After all, we have plenty of college prob/stat courses that are non calculus based. And I have personally been involved in Six Sigma training and the like in a manufacturing environment, where intuitive and phenomenological training was done, sometimes even with operators, but even at the professional level in a very "no limits needed" environment.

Of course there is the issue of what it displaces, but that's a tricky topic. I can think of Peggy Sue Got Married, where she tells her teacher that she knows from the future that she really won't ever need algebra! Clearly there is a bit of a fad/trend to play with the stats more, earlier. I kind of like it...since many kids will never get to calculus or be STEM superstars and this gives them a quick exposure to intuitive probs/stats so they have some better understanding of things in the newspaper.

1. How?

This is not a de novo problem. You/she can Google and find different resources.

For one thing, there are some nice (non rigorous) videos on the topic.

I would start by looking at existing approaches versus speculating on how to do it.

But with the proverbial gun at my head, I think the method she has is OK, but I would even just run a class exercise with 4 and then with 10 flips and then with 20 flips. Have the kids pair up, so they can socialize a little. And then do the histogram of class results on the board or with an overhead (same scale y scale as percentage, but align the endpoints of the x scale) and with the graphs superimposed or aligned above and below each other. I think after that, ask them what they think happens with 500 flips. Another provocation (US audience) is to ask if you have a sports team with a 55% chance of winning/game and it is single elimination (like Super Bowl last Sunday) or if it is best 4 out of 7, like the NBA. Which favors the better team more, which favors the lesser team more? (Don't prove the algebra, just let them debate it a little.)