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A random variable is a concept used in statistics to denote a variable that can take different outcomes randomly. In the context of our exercise, we define a random variable, denoted as \(X\), which represents the number of people with O-negative blood among a group of 10. Here, the random variable is discrete, which means it takes on a countable set of values—in this case, from 0 up to 10.
By identifying this random variable, we can perform various calculations to determine the likelihood of certain outcomes, such as the probability that at least one person in a group of 10 is a universal donor. This representation is key for making sense of probabilities in the scenario being considered.

The Bernoulli distribution is fundamental when dealing with individual trials where there are two outcomes: success or failure. In our context, 'success' refers to a person being a universal donor, while 'failure' denotes they are not.

• Each donor check (one person's blood type) is an independent trial.
• The probability of success (finding an O-negative person) is \( p = 0.072 \).
• The probability can be modeled using the formula for a single Bernoulli trial outcome.

When we expand to consider multiple trials, such as checking 10 people, the appropriate model becomes a Binomial distribution. This distribution extends the Bernoulli distribution for multiple trials.

Complementary probability is a useful strategy in probability calculations. It allows us to compute the probability of one event happening by reference to its complement - that is, the event not happening.
In this situation, rather than directly computing the probability of having at least one universal donor (our event of interest), we first consider the probability of having no universal donors at all, which is easier. The complement of "at least one" is "none."
The computation involves using the complement rule:

• Find the probability of no successes (universal donors), represented by \(P(X = 0)\).
• Subtract this from 1 to get the desired probability: \(P(X \geq 1) = 1 - P(X = 0)\).

This simplifies the computation and often reduces the number of calculations needed.

In probability calculations, especially with a Binomial distribution, several steps are typically involved, which include using formulas related to binomial coefficients and powers of probability terms.
To find the probability of zero universal donors, we calculate:

• Compute \( \binom{10}{0} = 1\) (this is the binomial coefficient for choosing 0 successes out of 10 trials).
• Multiply this by \((0.072)^0 (0.928)^{10}\), clarifying that we raise the probability of failure \((1 - p)\) to the number of trials \(n\).

Thus:\[P(X = 0) = 1 \times 0.928^{10}\].
Finally, we subtract this result from 1 to find the probability that there is at least one universal donor in the ten people: \(P(X \geq 1)\). With these steps, we can determine the likelihood of certain outcomes in any binomial scenario.