# Law of large numbers simulation dating, planning and Resources

## Law of Large Numbers and Simulations

## Law of large numbers

You used a die to simulate collecting the game pieces because of the cost involved in actually performing the experiment at the fast food restaurant. Maybe it just makes sense. With a single molecule, the motion appears to be quite random. For example, a fair coin toss is a Bernoulli trial. That is, what we expect should to happen.

Also, almost surely the ratio of the absolute difference to the number of flips will approach zero. But you could use a die very easy and convenient since it has six sides and keep track of how many of each game piece you received and how long it took you to get all six pieces. Initially, there are solute molecules on the left side of a barrier magenta line and none on the right.

Sample Assessment If you knew the sample space consisted of four outcomes, which of the following is true? With more molecules, there is clearly a trend where the solute fills the container more and more uniformly, but there are also random fluctuations. But now suppose you want to collect six game pieces being given away with every combo meal purchased at a fast food restaurant.

It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability. Students repeat an experiment a large number of times to explore the relationship between the relative frequency and the actual probability for an outcome. This is an example of a simulation. Sometimes a simulation is easy to carry out, as in the case of tossing the coin. What you have just done is conduct a simulation and found your own experimental probability.

What are the students doing? Let's think about rolling a single die. It could be very costly to keep going to the fast food restaurant until you collected all six. Intuitively, expected absolute difference grows, but at a slower rate than the number of flips, as the number of flips grows.

## Law of Large Numbers

What is the teacher doing? This is the theoretical probability. The relative frequency is how many times a certain color jellybean occurred divided by the total number of jellybeans drawn. The solute appears to move smoothly and systematically from high-concentration areas to low-concentration areas.

That is, the probability that the absolute difference is a small number, approaches zero as the number of flips becomes large. False The Big Idea Over the long run, macizo patagonico yahoo dating the relative frequencies of outcomes of chance processes stabilize as the sample size gets larger and can be used to estimate probabilities.

## Planning and Resources

Be sure students note that the distributions of results will often vary greatly when only a small number of samples is taken. It is what is supposed to happen. Lesson Snapshot Understanding The frequency is just the count for how many times a color was drawn.

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