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The geometric distribution models scenarios where you want to find the number of trials needed to get the first success in a series of independent experiments.
Each of these experiments has the same probability of success. This distribution is handy when you're waiting for a single specific outcome to occur for the first time.
For example, if you wanted to find how many people you need to examine to get just a single type O blood donor at a blood drive, the geometric distribution would be perfect.
It specifically answers questions like: *How many trials will it take to achieve the first success?* This assumes each individual trial is independent, meaning the outcome of one trial does not affect another, and the probability of success stays constant across each trial.
The probability distribution can be described by the probability mass function:

• \( P(X = k) = (1 - p)^{k-1}p \)

where:

• \(k\) is the trial number on which the first success occurs,
• \(p\) is the probability of success on each trial.

The negative binomial distribution is an extension of the geometric distribution.
Instead of finding when the first success occurs, the negative binomial models how many trials it will take to achieve a fixed number of successes.
In our exercise, we need to find out how many donors we have to examine to get three of them with type O blood. This situation perfectly fits the negative binomial distribution.
Here, you focus not just on the first success but on multiple successes, each independent of the others and with the same probability. It answers the question of how many trials you might need to achieve a set number of successful outcomes.
The negative binomial distribution has a probability mass function represented by:

• \( P(X = k) = \binom{k-1}{r-1} (1-p)^{k-r}p^r \)

where:

• \(k\) is the total number of trials needed to achieve \(r\) successes,
• \(p\) is the probability of success on each trial:
• \(r\) is the number of successful outcomes you are counting.

The expected value in probability distributions represents the average result you'd expect from many trials of a probabilistic process. It's an essential concept that gives you a measure of the center of distribution—a type of long-term average. For example, when using the negative binomial distribution, it can tell us on average how many people we'd need to examine to get a certain number of type O blood donors.
The formula to find the expected value \(E(X)\) of a negative binomial distribution when you want \(r\) successes, each success having a probability \(p\), is:

• \( E(X) = \frac{r}{p} \)

Using this formula, you simply plug in the number of successes you need and the probability of each success.
For the blood donor scenario, looking back at the solution, the expected value \( E(X) = \frac{3}{0.44} \) results in approximately 6.82, which tells us, on average, we should expect to examine about 7 people (after rounding) to find three type O blood donors. This formula allows you to anticipate the average number of trials needed in various settings where a certain number of successes is required.