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Probability is a measure of the likelihood that a specific event will occur. It is expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In the context of jury duty, each person who is called to serve has a certain probability of either being excused or available.
The probability that a person finds an excuse and cannot serve is given as 0.25. To find the probability that someone is available, we subtract this figure from 1, resulting in 0.75. This probability helps us calculate the likelihood of different outcomes, such as how many people will or will not be available when called.
Understanding these probabilities is crucial for calculating the expected number of jurors and evaluating how many individuals may need to be contacted to ensure a sufficient jury.

Expected value (often denoted as E(X)) is a key concept in probability and statistics. It represents the average outcome of a random event if it could be repeated many times. In this scenario, we're interested in how many of the 12 people called will typically be available for jury duty.
Using the formula for expected value in the context of binomial distribution, where we have a fixed number of trials, we multiply the total number of trials () by the probability of success (here, being available for jury duty).
The expected number of people available is calculated as: \[E(X) = n \times p = 12 \times 0.75 = 9\]This means, on average, 9 out of 12 people called will be available to serve on the jury. This expected value gives us an insight into what usually happens in a similar setting.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In simpler terms, it tells us how spread out the outcomes can be from the expected value.
For our jury duty example, standard deviation helps us understand the potential variability in how many people are actually available. We calculate it using the formula for standard deviation in a binomial distribution: \[\sigma = \sqrt{n \times p \times q}\]Plugging in our numbers: \[\sigma = \sqrt{12 \times 0.75 \times 0.25} = \sqrt{2.25} \approx 1.5\]This indicates that the number of people available could typically vary by about 1.5 people from the expected value (9), helping us understand the consistency of our outcome.

Binomial probabilities allow us to determine the likelihood of a specific number of successes in a fixed number of trials, given a constant probability of success. This is especially useful in scenarios like our jury duty problem.
Let's break down the calculation for finding specific probabilities. For instance, the probability that all 12 people are available, without any excuse, is calculated using:\[P(X = 0) = (0.75)^{12} \approx 0.0317\]Meaning there's a 3.17% chance that everyone will serve.
To find the probability of 6 or more people not being available, we first calculate the probability of less than 6 people finding excuses, then subtract it from 1 for the complement probability:\[P(X < 6) = \sum_{k=0}^{5} \binom{12}{k} (0.25)^k (0.75)^{12-k}\]Using the above sum, we compute the complement to find that about 36.6% of the time, 6 or more will be excused, demonstrating the usefulness of binomial probabilities in predicting real-world outcomes.