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When solving problems related to binomial distributions, such as the likelihood of a certain number of people supporting marijuana legalization, we often use the concept of cumulative binomial probability. It serves as a handy way to calculate the total probability of obtaining a certain number of successes or fewer in a series of trials.

The cumulative binomial probability answers questions of 'at most' or 'less than' type, by summing up the probabilities of all possible outcomes up to a specified point. This is particularly useful when dealing with a large number of trials where calculating individual probabilities would be impractical.

For example, in the provided exercise, instead of calculating the probability for each scenario where 0 to 110 people support legalization individually, we aggregate the probabilities. This cumulative approach harnesses the formula: \[\begin{equation} P(X \leq k) = \sum_{x=0}^{k} C(n, x) \cdot (p^x) \cdot ((1-p)^{n-x}) \end{equation}\]where \(P(X \leq k)\) is the cumulative probability of obtaining \(k\) or fewer successes. In this context, \(p\) represents the probability of an individual supporting marijuana legalization, and \(n\) is the total number of individuals sampled.

By using this cumulative approach, we answer part (a) of the exercise, finding the probability that at most 110 people out of 150 sampled support marijuana legalization.

Probability distribution is a core concept in statistics that outlines how probabilities are assigned to different possible outcomes of a random variable. It essentially tells us what the likelihood is of each outcome occurring. When dealing with discrete random variables, such as the number of people supporting legalization in our exercise, we use the binomial probability distribution.

The binomial distribution has only two possible outcomes for each trial, often termed as 'success' and 'failure'. In our scenario, a 'success' would be an individual who supports legalization, while a 'failure' would be one who doesn't. It characterizes the likelihood of achieving a certain number of successes in a number of trials, given the probability of a success in a single trial.

For example, in part (b) of the exercise, we are looking to calculate the probability for a specific range of outcomes - that between 90 and 110 people support legalization. This requires us to manipulate the cumulative distribution by subtracting the probability of fewer than 90 supporters from the probability of at most 110 supporters, effectively zooming in on the chances for our desired range of outcomes.

The expected value is a prediction of the average outcome if we were to repeat a random trial many times. In terms of our exercise, it's the average number of Americans we'd predict to support marijuana legalization in many samples each of 150 people. Mathematically, the expected value for a binomial distribution is calculated as the product of the number of trials \(n\) and the probability of success \(p\).\[\begin{equation} \mu = np \end{equation}\]This is part of Step 3 in our solution, where we have a sample size of 150 and a given probability of a single person supporting legalization at 64%. Multiplying these together gives us the expected number of people supporting legalization.

But there's also variability in any distribution, which is represented as the standard deviation. It measures how much the values in a set of data are likely to differ from the expected value. We calculate it for a binomial distribution using the formula: \[\begin{equation} \sigma = \sqrt{np(1-p)} \end{equation}\]Which gives us an idea of the spread or dispersion from the expected number of supporters, essentially allowing us to 'give or take' from the expected value to build a range of probable outcomes in our prediction.