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Law of Large Numbers and Simulations
In many cases you run across dealing with probability, percentages or probabilities are already given to you or are simple enough for you to compute on your own.
 
But how do you know the probability of tossing a coin and getting a tail is 1/2? Maybe someone told you. Maybe it just makes sense. After all there are only 2 possibilities and tails is one of them.
 
When talking about the probability of tossing a coin and getting a tail being 1/2 we are referring to the theoretical probability. That is, what we expect should to happen. Theoretically or in a perfect world, if we toss a coin 10 times, we would expect 5 of those tosses to be tails. 5/10 = 1/2. However, we don’t always live in a perfect world.
 
Find a coin and toss it 10 times. What happened? Did you get 1, 2, 5, 7, 9 tails? Take whatever number you got and put it over 10. So for example, say you got 7 tails. Then YOUR probability of getting tails is 7/10. What you have just done is conduct a simulation and found your own experimental probability.
 
A simulation is a method used to collect data dealing with probability. Sometimes a simulation is easy to carry out, as in the case of tossing the coin. But now suppose you want to collect six game pieces being given away with every combo meal purchased at a fast food restaurant. It could be very costly to keep going to the fast food restaurant until you collected all six. And there would be no guarantee that you’d actually get all six. 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. This is an example of a simulation. 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.
 
Let's think about rolling a single die. The probability of rolling any of the numbers is 1/6. This is the theoretical probability. It is what is supposed to happen. So that is we rolled a die 6 times, theoretically we should get a 1, 2, 3, 4, 5, and 6. And if we rolled a die 60 times, theoretically we should get 1, 2, 3, 4, 5, and 6 to each occur 10 times. Look at what happened when I rolled a die 60 times. I organized all the information in the chart below.
 
# rolled
tally
total
1 | | | | | | | | | | | | | | 14
2 | | | | | | | | | | 10
3 | | | | | | 6
4 | | | | | | | | 8
5 | | | | | | | | 8
6 | | | | | | | | | | | | | | 14
 
Theoretically I should have gotten 10 of each number but that didn't happen. Apparently I don't live in a perfect world where everything works out the way it is supposed to. My simulation didn't give me the theoretical probability. However it did give me experimental probabilities for my simulation or experiment.
 
Since there were a total of 60 rolls of the die, I can compute each experimental probability and it is shown in the table below.
 
# rolled
tally
total probability
1 | | | | | | | | | | | | | | 14 14/60 = 7/30
2 | | | | | | | | | | 10 10/60 = 1/6
3 | | | | | | 6 6/60 = 1/10
4 | | | | | | | | 8 8/60 = 2/15
5 | | | | | | | | 8 8/60 = 2/15
6 | | | | | | | | | | | | | | 14 14/60 = 7/30
 
If I set aside the results of my first 60 tosses and rolled the die another 60 times, the chances are very good that I would get a different set of experimental probabilities.
 
However, if everyone in a class of 30 students rolled the die 60 times for a grand total of 1800 rolls of the die and we combined everyone's results, our experimental probabilities would start to come very close to the theoretical probabilities that we expect. This is known as the Law of Large Numbers.
 
Simply stated, the Law of Large Numbers says that the more times you do something, the closer you will get to what is supposed to happen. In probability, this means that the more simulations we conduct the closer our experimental probability will get to the theoretical probability. In statistics, it means that the larger sample size you use the closer your sample will represent the entire population.
 
Since every simulation yields different results, there are no "Try These" problems for this lesson because it would be impossible to provide answers. However, you may wish to try rolling a die 60 times yourself to see what kind of results you might get. Keep in mind that yours should NOT look like the table in this lesson.



S Taylor

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