Law of Large Numbers

The Organic Chemistry Tutor
17 Sept 201906:14
EducationalLearning
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TLDRThe law of large numbers states that as the number of trials or samples increases, the observed average will approach the expected theoretical average. This concept is illustrated through examples of flipping coins and measuring weights. With more coin flips, the observed probability of heads approaches 0.5. And by measuring more people's weights in a town, the observed average weight converges between 161-162 pounds. So taking larger samples leads to more accurate estimates of a population's true parameters.

Takeaways
  • 😀 The law of large numbers states that as you increase the number of trials in an experiment, the observed average will approach the theoretical average.
  • 😊 For example, when flipping a coin multiple times, the percentage of heads should approach 50% as you flip more coins.
  • 🤓 You can use the law of large numbers to estimate characteristics about a population by taking larger and larger samples.
  • 😎 For instance, you can estimate the average weight of people in a town by measuring more and more people.
  • 🧐 As the sample size increases, the observed average weight converges towards the true average.
  • 🤔 Selecting just one person may give an extreme weight that isn't representative.
  • 😇 But increasing the sample size gives a better estimate of the true average weight.
  • 🤠 The key is that increasing the number of trials or sample size makes the observed average closer to the expected.
  • 😉 Even though chance plays a role, extremes get less likely in larger samples.
  • 😃 So the more trials, flips, or people measured, the closer statistics will be to theoretical values.
Q & A

  • What is the law of large numbers?

    -The law of large numbers states that as the number of trials or experiments increases, the average of the results obtained gets closer and closer to the expected theoretical average or true mean.

  • Why does tossing more coins lead to the observed probability getting closer to 0.5?

    -There is a 50% probability of getting heads when tossing a fair coin. As more coins are tossed, the observed probability tends to get closer and closer to the true 50% probability according to the law of large numbers.

  • How can the law of large numbers be used to estimate population parameters?

    -By taking larger and larger samples from a population, the sample averages get closer and closer to the true population parameter according to the law of large numbers. This allows population parameters to be estimated from samples.

  • What was the observed probability after tossing 10 and 100 coins?

    -After tossing 10 coins, the observed probability was 0.4 or 40%. After tossing 100 coins, the observed probability was 0.49 or 49%.

  • Why is taking just one sample not enough to estimate the average weight?

    -A single sample may give an extreme outlier value for the weight. More samples are needed for the average weight to converge to the true population average weight.

  • What was the range of average weights as more people were sampled?

    -With samples of 5, 10, 50, 100, 200, 500, and 1000 people, the average weights ranged from 154 to 163 pounds, getting closer to 161-162 pounds with more samples.

  • What are the two possibilities when flipping a fair coin?

    -The two possible outcomes when flipping a fair coin are heads or tails, with a 50% probability of each on any given flip.

  • Why does the observed probability get closer to 0.5 as more coins are tossed?

    -According to the law of large numbers, as the sample size increases, the observed probability will approach the expected theoretical probability. For a fair coin it is 0.5 or 50% heads.

  • What is an example of an extreme weight value for one sample?

    -Examples of extreme weight values from one sample could be someone who is obese at 230 pounds or underweight at 110 pounds, unlike the town's average.

  • How many people were assumed to live in the town?

    -It was stated that there were 20,000 people assumed to be living in the town used as an example.

Outlines
00:00
📈 Explanation and Examples of Law of Large Numbers

This paragraph provides an explanation of the law of large numbers using an example with coin flips. As the number of coin flips increases, the observed probability of heads approaches the theoretical probability of 0.5. More examples are provided to show that as sample size increases, the sample average converges to the true population average.

05:01
📊 Using Law of Large Numbers to Estimate Population Parameters

This paragraph provides another example related to estimating the average weight of people in a town. As the sample size increases, the average weight converges to between 161-162 pounds. This demonstrates how the law of large numbers can be used to estimate population parameters from samples.

Mindmap
Keywords
💡probability
Probability refers to the likelihood of an event occurring. In the video, it is used in the context of coin flips - there is a 50% (0.5) probability of getting heads when flipping a coin. As more coins are flipped, the observed probability gets closer to the theoretical probability, illustrating the law of large numbers.
💡sample
A sample refers to a subset selected from a larger population. The video gives the example of measuring the weight of some individuals from a town to estimate the average weight of the entire town's population. As the sample size increases, the average weight converges to the true population average.
💡converge
As the number of trials or sample size increases, the observed average converges towards the expected theoretical value. This convergence illustrates the law of large numbers.
💡experiments
Experiments refer to trials like coin flips where outcomes are observed. By performing more repeated experiments, observed averages get closer to expected values.
💡trials
Trials are experiments performed multiple times. The law of large numbers examines what happens as the number of trials increases.
💡outliers
Outliers refer to extreme values that are not representative. The video mentions that by sampling just one person, you may get an outlier weight not reflecting the town's average.
💡parameters
Parameters refer to overall statistical descriptors of a population, like an average weight. Samples allow estimating these population parameters.
💡observed average
The average result empirically measured from experiment outcomes or samples. It approaches the theoretical average.
💡theoretical average
The expected average based on probability calculations. As trials increase, the observed average converges to it.
💡sample size
The number of items selected in a sample from a larger population. As this increases, observed averages approach population averages.
Highlights

The law of large numbers states that as you increase the number of trials, the observed probability approaches the theoretical probability.

Flipping a fair coin has a 50% (0.5) chance of getting heads. But in a small number of trials you may not observe exactly 0.5.

As you flip more coins, the observed probability gets closer and closer to the theoretical 0.5 probability.

The more trials performed, the closer the observed results match the expected theoretical results. This is the essence of the law of large numbers.

You can use the law of large numbers to estimate characteristics of a whole population by taking samples.

With small samples, the average can vary significantly from sample to sample.

As the sample size increases, the average converges towards the true population average.

Large samples better approximate the actual population average than small samples.

Selecting just one individual could give an extreme outlier that misrepresents the population.

The more trials performed, the closer the observed probability gets to the expected theoretical probability.

The law of large numbers allows estimating unknown population parameters from samples.

Small samples lead to more variability in estimates compared to large samples.

As sample size grows, sample averages converge to the true population parameter.

Large samples provide better estimates of population characteristics than small samples.

Single observations may be unrepresentative extremes that skew results.

Transcripts

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