Blood Types of Ethnic Groups

This probability table concerns the blood types of three ethnic groups.

  Type O Type A Type BType AB
Group X.082.106.008.004
Group Y.135.141.018.006
Group Z.215.200.065.020
total.432.447.091.030

If someone is in Group Z, what is the probability they have blood type A? We want to compute P(A | Z). By Bayes formula,

P(A | Z) = P(A ∩ Z)/P(Z).

Clearly: P(A)= .447,   P(Z)=.5,   P(A ∩ Z)=.2

so   P(A | Z) = P(A ∩ Z)/P(Z) = .2/.5 = 40%

Similarly,

  P(Z | A) = P(Z ∩ A)/P(A) = .2/.447 ≈ .447

If an individual does not have blood type B, what is the probability this individual is in Group X?

We want: P(X | -B) = P(X ∩ -B)/P(-B). But

P(-B)= 1 - P(B) = 1 - .091 = .909

and P(X ∩ -B) = 0.82 + .106 + .004 = .192

so: P(X | -B) = .192/.909 ≈ .211 = 21.1%