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In the context of "probability distribution," the focus is on determining the likelihood of various outcomes when an experiment is performed. When we talk about probability distribution, we're dealing with the full range of possibilities and their associated probabilities. For Ray Allen's free throw example, the distribution of outcomes can be described using a binomial distribution.

A binomial distribution applies to scenarios where there are a fixed number of independent trials, each with only two possible outcomes (success or failure). In this situation, the number of successful shots, Ray makes, represents a random variable, which can take on discrete values of 0, 1, or 2 for two attempts.

• 0 successful shots: Probability is computed with the formula
• \( P(X = 0) = C(2, 0)(0.881)^0(1-0.881)^2 = 0.014 \)
• 1 successful shot: Calculated by
• \( P(X = 1) = C(2, 1)(0.881)^1(1-0.881)^1 = 0.208 \)
• 2 successful shots: The probability
• \( P(X = 2) = C(2, 2)(0.881)^2(1-0.881)^0 = 0.778 \)

This distribution provides the complete scenario of Ray's potential outcomes in terms of his free throw performance based on the given probabilities.

The "expected value" or mean of a random variable gives us a sense of the 'average' or typical outcome one would expect over numerous repetitions of the experiment.
In the case of a binomial distribution like Ray Allen's free throws, the expected value \(\mu\) is computed using the formula \(\mu = n \times p\). Here, \(n\) is the number of trials (shots), and \(p\) is the probability of success for each trial.

In this scenario:

• Number of trials \(n\): 2 attempts
• Probability of success \(p\): 0.881
• Expected value \(\mu\): \( 2 \times 0.881 = 1.762 \)

Thus, the mean number of successful shots Ray Allen would have over a large number of two-shot trials is approximately 1.762. This number represents an average outlook, showing what we can expect when looking at several instances of Ray taking two shots, even though he can't, in reality, make 1.762 shots.

The "binomial probability formula" is a key component in computing the probabilities for a binomial distribution. It's essential when you want to determine the likelihood of achieving a specific number of successful outcomes in a series of trials.

The general formula is:
\[ P(x) = C(n, x) \cdot p^x \cdot (1-p)^{n-x} \]

• \(C(n, x)\): The combination function, \(\binom{n}{x}\), counts how many ways you can choose \(x\) successful trials from \(n\) attempts.
• \(p^x\): The probability of success raised to the power of \(x\) indicates the likelihood of \(x\) successes.
• \((1-p)^{n-x}\): The probability of failure raised to the number of failures, \((n-x)\), shows the likelihood of everything else being unsuccessful.

Using this formula in Ray Allen's context helps calculate the probability of making 0, 1, or 2 shots out of 2 attempts. This provides a mathematical way of understanding how likely different outcomes can be, taking into account both success and failure possibilities on each try.