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A universal donor is someone whose blood can be used in transfusions to individuals with any blood type. This is particularly important in medical emergencies when there is no time to match the donor's blood type to the recipient's.

People with O-negative blood type are universal donors. This is because their blood lacks A, B, and Rh antigens, which are the primary triggers for immune reactions in blood transfusions. As a result, O-negative blood is considered compatible with any other type. However, only about 8% of the population has O-negative blood, why they are especially valued as donors.

Understanding who is a universal donor can be crucial in various medical contexts. It ensures anyone, regardless of blood type, can receive life-saving blood transfusions when needed.

In probability, independent events are events where the occurrence of one event does not affect the probability of the other.

For instance, when considering the probability of two people giving blood, each person's blood type is independent of the other. If each person has a 0.08 probability of having O-negative blood, we compute the probability for two people by multiplying these individual probabilities.

• For both individuals to have O-negative blood, the probability is calculated as follows: \(0.08 \times 0.08 = 0.0064\).
• Similarly, the probability that neither has O-negative blood is \((1 - 0.08) \times (1 - 0.08) = 0.92 \times 0.92 = 0.8464\).

These calculations demonstrate how outcomes are multiplied when events are independent, offering insights into more complex probability scenarios.

The complement rule is a fundamental principle in probability. It helps to calculate the probability of an event occurring by using the probability of its complement, or opposite.

Simply put, the probability of an event not happening is equal to one minus the probability of it happening. For instance, if the probability of a person having O-negative blood is 0.08, then the probability of not having it is \(1 - 0.08 = 0.92\).

This rule is beneficial for quickly determining non-occurrence probabilities. For example, to find the probability that at least one person in a group of people has a specific characteristic, we can subtract the probability that no members have that characteristic from one. In parts b(ii) and c of the exercise, this rule allows for efficient calculation of complex scenarios.

Probability calculations are pivotal to predicting the likelihood of events. They rely on understanding basic principles such as multiplication rules and complements.

To solve various parts of the problem, we use these principles:

• For b(i), the probability of both independently appearing individuals having O-negative is a simple multiply: \(0.08 \times 0.08 = 0.0064\).
• For b(ii), the probability of at least one having O-negative is \(1 - 0.8464\), calculated using the complement of neither having it, giving \(0.1536\).
• For b(iii), calculations for only one having O-negative involves addition of probabilities from two scenarios yielding \(0.1472\).
• For c, involving eight individuals, we extend the complement approach: \(1 - (0.92)^8 = 0.4868\).

These calculations emphasize the practical use of probability rules in tackling problems involving multiple scenarios and highlight the incremental changes as more variables are introduced.