The four major blood types are present in approximately thefollowing proportions

Question

The four major blood types are present in approximately thefollowing proportions in the population of the U.S.A.
Type:    A         B    AB    O
Proportion:    42%     10%   4%         44%

Note that each person's blood is exactly one of these four types.Type AB is a separate type, not the intersection of type A and typeB.

a) If two people are picked at random from this population, what isthe chance that their blood is of the same type? Of differenttypes?
b) If four people are picked at random, let P(k) be the chance thatthere are exactly k different blood types among them. Find P(k) fork = 1,2,3,4,.

Explanation / Answer

qa
P[both same type] = P[A-A] + P[B-B] + P[AB-AB] + P[O-O]
= 0.42^2 + 0.1^2 + 0.04^2 + 0.44^2 = 0.3816 <--------
P[both different] = 1 - 0.3816 = 0.6184 <-------

qb
P[k=1] = 0.42^4 + 0.1^4 + 0.04^4 + 0.44^4
P[k= 2] = DIY as explained at the end
P[k=3] = DIY as explained at the end
P[k=4] = 0.42*0.1*0.04*0.44 *4! perms

k = 3
-------
there can be 4c1*3c2=12 combos of types with 4!/2!=6 perms for each
eg A-A-B-AB is one combo with 6 perms,
Pr = 6 *0.42

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