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Imagine you are a basketball player. On average, out of free throw attempts, you successfully make shots. Well, this number can be called your expected value or expectation of success.
Simply put, expected value is the average value we expect to occur from an experiment if that experiment is repeated many times under the same conditions. This doesn't mean you will definitely get that result every time, but it's a prediction of the long-term average.
This concept is very useful in binomial distribution. Remember, binomial distribution is used for experiments that have only two possible outcomes (success or failure) and are performed repeatedly. Binomial expected value helps us predict how many successes we're most likely to get.
Binomial distribution has an expected value:
where is the number of trials and is the probability of success.
This formula is very intuitive. If the probability of success in one trial is , then in trials, the expected number of successes is:
You might be curious, why is the formula so simple? Let's break down the logic.
Each trial in a binomial distribution can be thought of as a small random variable, let's call it . This variable has a value of if the -th trial is successful, and if it fails.
The expected value for a single trial is:
Since the total number of successes () is the sum of all successes in each trial, then:
Using the properties of expected value, we can sum all expected values from each trial:
See? The total expected value is the product of the number of trials and the probability of success.
Let's apply this concept to an example.
Problem:
A fair die is rolled . What is the expected value for getting a five on the die from all the rolls?
Solution:
First, let's make sure this meets the requirements for binomial distribution:
Now we identify the parameters:
Now we can directly calculate the expected value using the formula:
So, what does this number mean?
Expected value doesn't mean you will get the number in one experiment (because that's impossible). What it means is: if you repeat this experiment of rolling a die hundreds or thousands of times, then on average you will get the number about per set of rolls.
In practice, in one set of rolls, you might get the number as many as , , , or even . But when averaged over the long term, the result will approach .
