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Chapter 5: Problem 91

Universal blood donors People with type O-negative blood are universal donors. That is, any patient can receive a transfusion of O-negative blood. Only \(7.2 \%\) of the American population have O-negative blood. If we choose 10 Americans at random who gave blood, what is the probability that at least 1 of them is a universal donor?

Short Answer

Expert verified
The probability is approximately 0.487, or 48.7%.

Step by step solution

01

Define the Problem

We want to find the probability that at least one person out of 10 randomly chosen people has O-negative blood (universal donor). This is equivalent to asking for the complementary probability of none having O-negative blood.
02

Identify the Complementary Probability

Instead of calculating the probability that at least one person is a universal donor, calculate the probability that none of the 10 people is a universal donor. The probability that one person does not have O-negative blood is \(1 - 0.072 = 0.928\).
03

Calculate the Probability of None Being a Universal Donor

The probability that all 10 people do not have O-negative blood is \(0.928^{10}\).
04

Compute Complementary Probability for At Least One Universal Donor

Subtract the probability that none of the people has O-negative blood from 1, to find the probability that at least one does: \(1 - 0.928^{10}\).
05

Perform the Calculation

Using a calculator, compute \(0.928^{10} \approx 0.513\) and subtract from 1: \(1 - 0.513 = 0.487\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complementary Probability
In probability theory, sometimes it is easier to calculate the probability of the opposite event to the one you are interested in. This approach is known as complementary probability. When you're trying to find the probability of an event occurring, you can instead find the probability of it not occurring and subtract it from 1.

  • The concept hinges on the fundamental principle that the total probability of all possible outcomes of a random experiment is 1.
  • For example, if you want to know the probability of an event happening at least once, it can be simpler to calculate the probability of it not happening at all and then subtract from 1.
In this exercise, to find the probability that at least one of the 10 randomly chosen Americans has O-negative blood, it is easier to first calculate the probability that none of them is a universal donor. Since complementary events are mutually exclusive and collectively exhaustive, this method is straightforward and efficient.
Universal Donor
People with type O-negative blood are often referred to as universal donors. This is because their blood can be transfused to any patient, regardless of the recipient's blood type. It's an extremely valuable and crucial type in medical emergencies and surgeries.

  • Universal donors have O-negative blood, which lacks any antigens that could provoke an immune response in the recipient.
  • Despite being universally compatible, only a small fraction of the population鈥攋ust about 7.2% in the US鈥攑ossesses this blood type.
Understanding universal donors is important in contexts such as blood donation and transfusion. Since only a small percentage of people have this blood type, there is always a high demand for donations from individuals with O-negative blood.
Random Selection
Random selection plays a key role in probability and various fields such as statistics, science, and research. When individuals or items are chosen at random, it means each has an equal chance of being selected with no bias influencing the choice.

  • This concept reinforces fairness and ensures that every possible outcome has a chance to occur.
  • In probability exercises, random selection is critical because it guarantees the validity of the calculations by aligning them with the principles of equal likelihood.
In the exercise given here, the 10 Americans are chosen randomly to demonstrate how likely it is that at least one person among them will have O-negative blood. By ensuring the selection is random, we model real-life scenarios accurately, capturing genuine randomness inherent in such biological distributions.

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