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The expected value is a fundamental concept in probability and statistics that provides a measure of the center, or average outcome, of a random variable. It's essentially what you would anticipate as the average outcome if you could repeat an experiment an infinite number of times.

In the context of a binomial distribution, the expected value is calculated by multiplying the number of trials (the sample size) by the probability of success in a single trial. In our exercise, with a sample size of 200 millennials and a success probability of 53% using a library, the expected value, denoted as E, is calculated as follows:
\[ E = n \times p = 200 \times 0.53 = 106 \]
This indicates that, in a typical group of 200 millennials, one would expect around 106 individuals to have used a library or bookmobile within the last year.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the situation of a binomial distribution, it helps us understand how much the number of successes can differ from the expected value.

To calculate the standard deviation (\( \(\sigma\) \)) for a binomial distribution, you use the square root of the product of the sample size (n), the probability of success (p), and the probability of failure (1-p). Following our exercise:
\[ \sigma = \sqrt{n \times p \times (1-p)} = \sqrt{200 \times 0.53 \times (1 - 0.53)} = 7.14 \]
This value gives us a sense of the 'give or take' around our expected value — in this case, about 7 millennials. Standard deviation is essential because it allows us to set expectations about how much variation we might see in the real world.

Sample size is the number of observations in a sample and is directly proportional to the accuracy of the estimates we can make about a population. In statistical terms, the larger the sample size, the smaller the standard deviation, which implies that our estimates are more likely to be near the true population value.

When considering sample size in the context of binomial distribution, it's important to remember that it affects both the expected value and the standard deviation. For instance, changing the sample size in our exercise would alter both the expected number of millennials using a library and the expected variation (standard deviation) around that number. As such, choosing an appropriate sample size is critical for achieving reliable statistical results.

Probability is a way of quantifying the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 represents certainty. In the context of a binomial distribution, the probability pertains to the success of a single trial.

For our exercise, the probability that a millennial has used a library in the past year is 53%, or 0.53 when expressed as a decimal. This probability is key to determining the expected value and standard deviation. It defines the underlying likelihood, which is then scaled up by the sample size to predict outcomes for larger groups, as well as calculating the range of possible outcomes around the expected value.