Question

The American Red Cross says that about \(45 \%\) of the U.S. population has Type O blood, \(40 \%\) Type A, \(11 \%\) Type \(\mathrm{B}\), and the rest Type \(\mathrm{AB}\). a) Someone volunteers to give blood. What is the probability that this donor 1) has Type AB blood? 2) has Type A or Type B? 3) is not Type \(\mathrm{O}\) ? b) Among four potential donors, what is the probability that 1) all are Type \(\mathrm{O}\) ? 2) no one is Type \(\mathrm{AB}\) ? 3) they are not all Type \(\mathrm{A}\) ? 4) at least one person is Type B?

Short answer

a) 1) 0.04, 2) 0.51, 3) 0.55; b) 1) 0.041, 2) 0.849, 3) 0.9744, 4) 0.3726.

Step-by-step solution

Determine Type AB Probability

To find the probability that a random blood donor is Type AB, we first need to determine the proportion of the U.S. population with Type AB blood. According to the problem, 45% are Type O, 40% are Type A, and 11% are Type B, which sums up to 96%. Since these are the only types specified that do not include AB, the remaining percentage must be for Type AB. So, the percentage of Type AB is 100% - 96% = 4%. Thus, the probability that someone has Type AB blood is 0.04.

Probability of Type A or Type B

To find the probability of someone having Type A or Type B blood, add the probabilities of having Type A and having Type B. The probability for Type A is 0.40 and for Type B is 0.11. Therefore, the probability of having Type A or Type B blood is 0.40 + 0.11 = 0.51.

Probability of Not Being Type O

To find the probability that someone is not Type O, subtract the probability of being Type O from 1. The probability of being Type O is 0.45, therefore, the probability of not being Type O is 1 - 0.45 = 0.55.

All Donors Are Type O

To find the probability that all four potential donors are Type O, raise the probability of one Type O donor to the power of 4, since these are independent events. That is \( (0.45)^4 = 0.041\).

No Donors Are Type AB

To find the probability that no donors are Type AB, find the probability that one donor is not Type AB and raise it to the 4th power (again assuming independence). The probability that one donor is not Type AB is 1 - 0.04 = 0.96. Therefore, the probability that no one is Type AB is \( (0.96)^4 = 0.849\).

Not All Donors Are Type A

To find the probability that not all donors are Type A, first find the probability that all are Type A and subtract from 1. The probability that one is Type A is 0.40. Thus, the probability that all four are Type A is \( (0.40)^4 = 0.0256\). Therefore, the probability that not all are Type A is 1 - 0.0256 = 0.9744.

At Least One Donor is Type B

To find the probability that at least one donor is Type B, use the complement rule by first finding the probability that no one is Type B. The probability that one is not Type B is 1 - 0.11 = 0.89. Therefore, the probability that none are Type B is \( (0.89)^4 = 0.6274\). Then, the probability that at least one is Type B is 1 - 0.6274 = 0.3726.

Key concepts

Blood Type Distribution

Blood type distribution plays a crucial role in understanding the likelihood of different blood groups within a population. In this example, consider the U.S. population, where the American Red Cross has documented various percentages for each blood type:

• Type O: 45%
• Type A: 40%
• Type B: 11%
• Type AB: 4%

Together, these percentages should ideally sum up to 100%, representing all possible blood types. Knowing these proportions helps in calculating the probability of randomly selecting a person with a particular blood type. By understanding this distribution, we can easily compute probabilities to tackle various questions related to events such as those discussed in blood donation scenarios.

Complement Rule

The complement rule is a fundamental concept in probability that helps to simplify complex problems. When determining the probability of an event, it can often be easier to compute the chance of its complement – the event not happening – and subtract this from 1. For instance, if you want to find the probability that a blood donor does not have Type O blood, start with the known probability of having Type O, which is 0.45. The complement would be the probability that they are not Type O: \[ P( ext{not Type O}) = 1 - P( ext{Type O}) = 1 - 0.45 = 0.55 \] Using the complement rule often simplifies calculations, especially in cases involving multiple scenarios or large sample spaces. It's a convenient method when the direct calculation seems more challenging.

Independent Events

In probability, events are labeled as independent if the outcome of one event does not influence the outcome of another. This is a key concept, particularly useful in calculating probabilities when considering multiple events over a period.For example, consider four randomly selected blood donors. We can declare each donor's blood type as an independent event. This meaning, the blood type of one donor does not affect the blood type of another. If you're evaluating the probability that all four donors have Type O blood, you calculate this by raising the probability of a single Type O event to the power of the number of donors: \[ P( ext{all Type O}) = (0.45)^4 \] Understanding independence is crucial in analyzing real-world scenarios accurately, allowing correct aggregation of probabilities across events.

Combined Probabilities

Combining probabilities involves adding or multiplying probabilities to predict outcomes involving more than one separate event. The approach depends on whether the combined events are mutually exclusive or independent.Events are mutually exclusive if they cannot happen at the same time. For instance, a blood donor cannot simultaneously have Type A and Type B blood. In such cases, you add the probabilities to compute the combined probability of either event occurring: \[ P( ext{Type A or Type B}) = P( ext{Type A}) + P( ext{Type B}) = 0.40 + 0.11 = 0.51 \] For independent events, the rule is to multiply their probabilities. This would be the case for calculating the probability of all donors being a specific blood type, as seen in the probability of all donors being Type O: \[ P( ext{all Type O}) = (0.45)^4 \] Learning to apply the correct operations based on the relationship between events is essential for accurate probability calculations.