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Chapter 1: Q14E (page 21)

Consider, once again, the four blood types A, B, AB, and O described in Exercise 8 in Sec. 1.4 together with the two antigens anti-A and anti-B. Suppose that, for a given person, the probability of type O blood is 0.5, the probability of type A blood is 0.34, and the probability of

type B blood is 0.12.

a. Find the probability that each of the antigens will

react with this person鈥檚 blood.

b. Find the probability that both antigens will react with

this person鈥檚 blood

Short Answer

Expert verified

a.

The probability that antigen-A will react with this person鈥檚 blood is: 038

The probability that antigen-B will react with this person鈥檚 blood is: 0.16

b.

The probability that both antigens will react with this person鈥檚 blood is: 0.04

Step by step solution

01

Given information

Let A be the event that antigen-A will react with the given person鈥檚 blood.

Let B be the event that antigen-B will react with the given person鈥檚 blood

The probability of type O blood is:

\(\Pr \left( {Otype} \right) = 0.5\)

The probability of type A blood is:

\(\Pr \left( {Atype} \right) = 0.34\)

The probability of type B blood is:

\(\Pr \left( {Otype} \right) = 0.12\)

Therefore, the probability of AB blood type is:

\(\Pr \left( {ABtype} \right) = 0.04\)

02

Computing the probability

a)

The probability that antigen-A will react with this person鈥檚 blood is:

\(\begin{aligned}{}\Pr \left( A \right) &= \Pr \left( {Atype} \right) + \Pr \left( {ABtype} \right)\\ &= 0.34 + 0.04\\ &= 0.38\end{aligned}\)

Since antigen-A reacts with blood type A or blood type AB,

The probability that antigen-A will react with this person鈥檚 blood is:

\(\Pr \left( A \right) = 0.38\)

The probability that antigen-B will react with this person鈥檚 blood is:

\(\begin{aligned}{}\Pr \left( A \right) &= \Pr \left( {Btype} \right) + \Pr \left( {ABtype} \right)\\ &= 0.12 + 0.04\\ &= 0.16\end{aligned}\)

Since antigen-B reacts with blood type B or blood type AB,

The probability that antigen-B will react with this person鈥檚 blood is:

\(\Pr \left( B \right) = 0.16\)

03

Finding the probability  

b.

The probability that both antigens will react with this person鈥檚 blood is:

\(\begin{aligned}{}\Pr \left( {A \cap B} \right) &= \Pr \left( {ABtype} \right)\\\Pr \left( {A \cap B} \right) &= 0.04\end{aligned}\)

Antigen-A and antigen-B both react with blood type AB only,

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