Now you will learn about relative frequency, which is closely related to the law of large numbers.
You have previously learned that the probability of getting a five when you roll a die is
If you roll a die 12 times, you may not get a five exactly one sixth of the time. After 12 throws that would be
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times. You may get five fives, three fives, or maybe no fives.
To get an answer as to why we say the probability of getting a five when you roll a die is
Imagine rolling a die 12 times and writing down how many fives you get. The relative frequency is the ratio of how many fives you got (number of favorable outcomes), to how many times you rolled the die (number of trials). If you got three fives, the relative frequency is
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This is a lot more than you expected, which was
The relative frequency of getting a five when rolling a die does not need to be the same as the probability (
Note! The relative frequency approaches the probability when the number of trials becomes very large!
Theory
The relative frequency is the ratio between how many times an event occurs
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The average age of people in the US is about 38 years old. You decide to personally ask a bunch of people how old they are. After you ask each person, you calculate the average age of the people you have asked so far. How many people do you have to ask before you get 38 years old as the average? Maybe it’s sometimes sufficient to ask five people. But what if the five you asked are all a group of friends in junior high? Then you will certainly not find the average age of the population to be 38 years old.
It makes sense that the answer is 38 if you just ask enough people. But how many are enough?
This is what the law of large numbers has something to say about.
Rule
If you perform an experiment with a large number of trials, the results of the experiment will approach the expected value of the experiment.
Unless you are investigating something very unlikely, then somewhere between
Below, I made a graph of the relative frequency of getting a five when rolling a die. The
You can see from the figure that after approximately
Next, let’s look at some examples where it’s important to know the law of large numbers.
Example 1
You decide to try your luck at gambling. There’s a
Example 2
I think it is wise to insure my house, because I’m afraid that something bad will happen to it. For example, a fire or another accident beyond my control may occur. The insurance company must know how likely it is that damage will occur in order to calculate how much my insurance will cost. If my house burns down, the insurance company will have to pay me a lot of money. I have to pay a much smaller monthly amount to the insurance company for full coverage in the event this happens.
Let’s say there’s a
Rolling a die
To avoid having to roll a die
Simulating means that we are imitating an experiment from reality on a computer. We can mimic dice rolls on the computer, rolling a virtual die many, many times over a short period of time. In this way, you’re then able to see what the probability of an event is, with the computer’s help. The computer calculates the relative frequency for such a large
Computers are a very important tool when working with probability.