56 ELEMENTARY PROBABILITY Ch. 3 



— assuming that an ace was drawn on the first draw — there are 3 

 aces among 51 cards remaining in the deck. Hence, P(E 2 /E l ) = 

 3/51; then, by formula 3.17, P(A, A) = (4/52) • (3/51) = 1/221. 



There are many situations in which those chance occurrences 

 which would be considered as the single events do not occur with 

 equal relative frequencies. For example, a coin may be biased so 

 that the heads side turns up more frequently than the tails side. Un- 

 der such conditions, we cannot assign a probability of 1/2 to the 

 occurrence of heads (and likewise for tails) and employ the simple 

 arguments used above. However, we can think of determining the 

 appropriate probabilities for these single events by empirical means, 

 that is, by many actual trials under the specified conditions. For 

 example, we could toss the coin in question many times and then use 

 the observed proportion of heads as an approximation to the true 

 probability, p. Thereafter, formulas 3.15, 3.16, and 3.17 can be used. 



An interesting and instructive application of the probability meth- 

 ods introduced above can be made to the study of human blood 

 groups. If the red blood corpuscles of one individual's blood are 

 mixed with the blood serum of another person (as in transfusions), 

 one of two general results will be observed to follow: the red cor- 

 puscles will disperse evenly through the recipient's blood as though 

 in their own serum, or they will form clumps of cells. The latter 

 reaction is called agglutination, and it is so undesirable that there is 

 considerable interest in preventing its occurrence. To that end, 

 bloods are classified according to certain systems. One such system 

 is based on the known existence of factors A and B each (or both) 

 of which may be either present or absent from any person's blood. 

 The following four blood groups are based on the A and B factors: 



(1) type 0: neither A nor B present in the blood; 



(2) type A: factor A present but not factor B; 



(3) type B: factor B present but not factor A; 



(4) type AB: both of the factors A and B are present in the blood. 



There are several interesting features about the A and B factors 

 in blood. (1) They are inherited essentially in accord with simple 

 Mendelian laws of inheritance, a circumstance which requires meas- 

 ures of probability. (2) Various racial or geographic groups tend to 

 differ from each other in the proportions carrying the A and/or B 

 factors, thus providing a source of some additional evidence about 

 racial origins. (3) The mode of inheritance of the A-B groups can 

 be used in genetic studies and in some legal problems. 



