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8: Law of Large Numbers

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    • 8.1: Discrete Random Variables
      We are now in a position to prove our first fundamental theorem of probability. We have seen that an intuitive way to view the probability of a certain outcome is as the frequency with which that outcome occurs in the long run, when the experiment is repeated a large number of times.
    • 8.2: Continuous Random Variables

    Thumbnail: Diffusion is an example of the law of large numbers. Initially, there are solute molecules on the left side of a barrier (magenta line) and none on the right. The barrier is removed, and the solute diffuses to fill the whole container. Top: With a single molecule, the motion appears to be quite random. Middle: With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations. Bottom: With an enormous number of solute molecules (too many to see), the randomness is essentially gone: The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas. In realistic situations, chemists can describe diffusion as a deterministic macroscopic phenomenon, despite its underlying random nature. (Pubic Domain; Sbyrnes321 via Wikipedia).

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