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Chapter 13: Problem 36

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 O? b) Among four potential donors, what is the probability that 1) all are Type O? 2) no one is Type AB? 3) they are not all Type A? 4) at least one person is Type B?

Short Answer

Expert verified
1) 0.04, 2) 0.51, 3) 0.55; b) 1) 0.041, 2) 0.8493, 3) 0.9744, 4) 0.3726.

Step by step solution

01

Calculate Probability of Type AB Blood

The percentage of the population with Type AB blood is the remainder after accounting for Type O, A, and B. We know: \[ \text{Type O} = 45\%, \text{Type A} = 40\%, \text{Type B} = 11\% \] Therefore, to find Type AB:\[ \text{Type AB} = 100\% - (45\% + 40\% + 11\%) = 4\% \]So the probability that a donor has Type AB blood is \(0.04\).
02

Calculate Probability of Type A or Type B

The probability of a donor having either Type A or Type B blood is the sum of the probabilities of each:\[ \text{Type A or B} = \text{P(Type A)} + \text{P(Type B)} = 0.40 + 0.11 = 0.51 \]Thus, the probability is \(0.51\).
03

Calculate Probability of Not Type O

The probability that a donor is not Type O is calculated by subtracting the probability of Type O from 1:\[ \text{Not Type O} = 1 - \text{P(Type O)} = 1 - 0.45 = 0.55 \]So the probability is \(0.55\).
04

Calculate Probability that All Four are Type O

Using the individual probability of a donor being Type O \((0.45)\), we find the probability that all four donors are Type O:\[ (0.45)^4 = 0.041 \]This results in a probability of \(0.041\).
05

Calculate Probability that No One is Type AB

The probability a single donor is not Type AB is \(1 - 0.04 = 0.96\). The probability that none of the four are Type AB is:\[ (0.96)^4 = 0.8493 \]This gives a probability of approximately \(0.8493\).
06

Calculate Probability that Not All are Type A

The probability that one person is Type A is \(0.40\), so the probability all are Type A is:\[ (0.40)^4 = 0.0256 \]Thus, the probability that not all are Type A is:\[ 1 - 0.0256 = 0.9744 \]So the probability is approximately \(0.9744\).
07

Calculate Probability that at Least One Person is Type B

The probability a single person is not Type B is \(1 - 0.11 = 0.89\), and therefore, the probability that all four are not Type B is:\[ (0.89)^4 = 0.6274 \]The probability of at least one being Type B is:\[ 1 - 0.6274 = 0.3726 \]So the probability is approximately \(0.3726\).

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Key Concepts

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

Blood Types
Blood types are categories of blood that are distinguished by the presence or absence of certain antigens on the surface of red blood cells. These antigens help determine compatibility for blood transfusions.

There are four main blood types in the ABO system:
  • Type O: Lacks A and B antigens.
  • Type A: Has A antigens.
  • Type B: Has B antigens.
  • Type AB: Has both A and B antigens.
In this exercise, the percentage distribution of these blood types in the U.S. is given, which is essential information for calculating various probabilities.
Events in Probability
An event in probability refers to a set of outcomes of an experiment or situation. For example, the event of someone having Type O blood means considering all outcomes where someone has this blood type.

Simple events are those with a single outcome, like a person having Type A blood. Composite events involve multiple outcomes, such as a person having either Type A or Type B blood. Understanding the type of event helps to define the range of outcomes to consider when calculating probabilities.
Multiple Events Probability
The concept of multiple events probability deals with scenarios where more than one outcome is considered simultaneously. For example, determining probabilities in sequences of events, like multiple people having certain blood types.

To assess probability with multiple events:
  • For independent events, multiply their individual probabilities. For instance, the probability of all four donors having Type O blood is \( (0.45)^4 \).
  • For events that are complementary, such as having a blood type that is not Type A, use the complement rule for more straightforward calculations.
These principles allow for more detailed probability estimates in complex scenarios, like a group of potential blood donors.
Complement Rule in Probability
The complement rule is a fundamental concept in probability that helps simplify complex probability calculations. The complement of an event is essentially the scenario in which the event does not occur.

Using the complement rule involves these steps:
  • Identify the probability of the event occurring \( P(A) \).
  • Calculate the probability of the complement of the event as \( 1 - P(A) \).
A typical use case in this exercise is determining the probability of not having Type O blood, knowing that \( 45\% \) have it. Therefore, \( P( ext{Not Type O}) = 1 - 0.45 = 0.55 \). This offers an efficient way to deduce probabilities without directly calculating from scratch.

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Most popular questions from this chapter

In Exercise 35 you calculated probabilities of getting various \(\mathrm{M} \& \mathrm{M}\) 's. Some of your answers depended on the assumption that the outcomes described were disjoint; that is, they could not both happen at the same time. Other answers depended on the assumption that the events were independent; that is, the occurrence of one of them doesn't affect the probability of the other. Do you understand the difference between disjoint and independent? a) If you draw one \(\mathrm{M} \& \mathrm{M},\) are the events of getting a red one and getting an orange one disjoint, independent, neither? b) If you draw two M\&M's one after the other, are the events of getting a red on the first and a red on the second disjoint, independent, or neither? c) Can disjoint events ever be independent? Explain.

A Pew Research poll in 2011 asked 2005 U.S. adults whether being a father today is harder than it was a generation ago. Here's how they responded: $$\begin{array}{lc}\text { Response } & \text { Number } \\ \text { Easier } & 501 \\ \text { Same } & 802 \\ \text { Harder } & 682 \\ \text { No Opinion } & 20 \\ \text { Total } & 2005\end{array}$$ If we select a respondent at random from this sample of 2005 adults, a) what is the probability that the selected person responded "Harder"? b) what is the probability that the person responded the "Same" or "Easier"?

According to Pew Research, the contact rate (probability of contacting a selected household) was \(90 \%\) in 1997 and \(62 \%\) in \(2012 .\) However, the cooperation rate (probability of someone at the contacted household agreeing to be interviewed) was \(43 \%\) in 1997 and dropped to \(14 \%\) in 2012 a) What was the probability (in 2012 ) of obtaining an interview with the next household on the sample list? (To obtain an interview, an interviewer must both contact the household and then get agreement for the interview.) b) Was it more likely to obtain an interview from a randomly selected household in 1997 or in \(2012 ?\)

Although it's hard to be definitive in classifying people as right- or left- handed, some studies suggest that about \(14 \%\) of people are left-handed. since \(0.14 \times 0.14=0.0196,\) the Multiplication Rule might suggest that there's about a \(2 \%\) chance that a brother and a sister are both lefties. What's wrong with that reasoning?

A certain bowler can bowl a strike \(70 \%\) of the time. What's the probability that she a) goes three consecutive frames without a strike? b) makes her first strike in the third frame? c) has at least one strike in the first three frames? d) bowls a perfect game ( 12 consecutive strikes)?
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