SOCR EduMaterials Activities LawOfLargeNumbers

This is a heterogeneous Activity that demonstrates the Law of Large Numbers (LNN)

Example

The average weight of 10 students from a class of 100 students is most likely closer to the real average weight of all 100 students, compared to the average weight of 3 randomly chosen students from that same class. This is because the sample of 10 is a larger number than the sample of only 3 and better represents the entire class. At the extreme, a sample of 99 of the 100 students will produce a sample average almost exactly the same as the average for all 100 students. On the other extreme, sampling a single student will be an extremely variant estimate of the overall class average weight.

Statement of the Law of Large Numbers

If an event of probability p is observed repeatedly during independent repetitions, the ratio of the observed frequency of that event to the total number of repetitions converges towards p as the number of repetitions becomes arbitrarily large.

SOCR Demonstrations of the LLN

• Exercise 1: Go to the SOCR Experiments and select the Binomial Coin Experiment. Select the number of coints (n=3) and probability of heads (p=0.5). Notice the blue model distribution of the Number of Heads (X), in the right panel. Try varying the probability (p) and/or the number of coins (n) and see how these parameters affect the shape of this distribution. Can you make sense of it? For example, if p increases, why does the distribution move to the right and become concentrated at the right end (i.e., left-skewed)? And vice-versa, if you decrease the probability of a head, the distribution will become skewed to the right and centered in the left end of the range of X (\(0\le X\le n\)).

Let us toss three coins 10 times (by clicking 10 times on the RUN button of the applet on the top). We observe the sampling distribution of X, how many times did we observe 0, 1, 2 or 3 heads in the 10 experiments (each experiment involves tossing 3 coins independently) in red color superimposed to the theoretical (exact) distribution of X, in blue. The four panels in the middle of the Binomial Coin Applet show:

Coin Box Panel, where all coin tosses are shown	The theoretical (blue) and sampling (observed, red) distributions of the Number of Heads in the series of 3-coin-toss experiments (X)
Summary statistics table that includes columns for the index of each Run, the Number of Heads and the Proportion of heads in each experiment	Numerical comparisons of the Theoretical and Sampling distribution (\(0\le X\le n\)) and two statistics (mean, SD)

Now take a snapshot of these results or store these summaries in the tables on the bottom.

According to the LLN, if we were to increase the number of coins we tossed at each experiments, say from n=3 to n=9, we need to get a better fit between theoretical and sampling distributions. Is this the case? Are the sample and theoretical (Binomial) probabilities less or more similar now (n=9), compared to the values we got when n=3?

Of course, we are doing random sampling, so nothings is guaranteed, unless we ran a large number of coin tosses (say > 50) which you can do by setting n=50 and pressing the Run button. How close to the theoretical p is now the empirical sample proportion of Heads (Column M)? These should be very close.

• Common Misconceptions regarding the LNN:
  • Misconception 1: If we observe a streak of 10 consecutive heads (when p=0.5, say) the odds of the \(11^{th}\) trial being a Head is > p! This is of course, incorrect, as the coin tosses are independent trials (an example of a memoryless process).
  • Misconception 2: If run large number of coin tosses, the number of heads and number of tails become more and more equal. This is incorrect, as the LLN only guarantees that the sample proportion of heads will converge to the true population proportion (the p parameter that we selected). In fact, the difference |Heads - Tails| diverges!

• Exercise 2: Much like we did above with coin tosses, one can see the action of the LLN in a variety of situations where one samples and looks for consistency of probabilities of various events (theoretically vs. empirically). Such examples may include Cards and Coins Experiments, Dice Experiments, etc.

Let's try the Ball and Urn Experiment. Go to the SOCR Experiments and select the this experiment in the drop-down list. Select N (population size), R (number of Red balls <=N) and n (sample-size, number of balls to draw from the urn. Note that you can sample with (Binomial) or without (Hypergeometric) replacement). Notice the blue model distribution of Y, the Number of Red balls in the sample of n balls, in the right panel. Again, in Red we see the sampling distribution of Y, as we do this experiment repeatedly. The probability of drawing a Red ball will depend on whether we replace the balls and the proportion of Red balls in the urn. For example, if R increases, the distribution moves to the right and become concentrated at the right end (i.e., left-skewed). Analogously, if you decrease R, the distribution will become skewed to the right and centered in the left end of the range of Y (\(0\le Y\le R\)).

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Try repeating what we did in the Coin Toss Exercise above and see the effects of the LLN in this situation (with respect to the sample size n).

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• SOCR Home page: http://www.socr.ucla.edu

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