EX 2.27 blood type

• Four individuals have responded to a request by a blood bank for blood donations.

• None of them has donated before,

  • so their blood types are unknown.

• Suppose

  • only type $\mathrm{O}+$ is desired
  • only one of the four actually has this type.

• If the potential donors are selected in random order for typing,

  • what is the probability that at least three individuals must be typed to obtain the desired type?

• Making the identification

  • $B$ = { first type not $\mathrm{O}+$ }
  • $A$ = { second type not $\mathrm{O}+$ },

• Then

  • $P\left(B\right)=3/4$

• Given that

  • the first type is not $\mathrm{O}+$ ,

• Then

  • two of the three individuals left are not $\mathrm{O}+$,

• so $P\left(A\mid B\right)=2/3$ .

• The multiplication rule now gives

$$\begin{array}{rl}& P\left(\text{at least three individuals are typed}\right)\\ =& P\left(A\cap B\right)\\ =& P\left(A\mid B\right)\cdot P\left(B\right)\\ =& \frac{2}{3}\cdot \frac{3}{4}=\frac{6}{12}\\ =& .5\end{array}$$