SOCR EduMaterials Activities LawOfLargeNumbers
Contents
SOCR Educational Materials - Activities - SOCR Law of Large Numbers Activity
This is a heterogeneous Activity that demonstrates the Law of large Numbers (LNN)
The Law of Large Numbers (LLN)
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 students 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.
Complete details about the LLN can be found here
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 do these parameters effect 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)? 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 theoredical (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 we re 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 emperical 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 consequtive 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 closer. 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!
- SOCR Home page: http://www.socr.ucla.edu
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