Sec. 3.2 PERMUTATIONS AND COMBINATIONS 63 



to change ttts assumption. (2) Similarly, for type A. (3) The probability that 

 a person who tests Rh+ is specifically Rh/rh is 2/3, and similarly for P+ if 

 there is no other information available which would change the probability. 



14. Suppose that a tortoise (land turtle) is wandering at random on a 50 by 

 50-foot lawn enclosed by a fence. He is equally likely to be on any particular 

 square foot of lawn one could designate in advance. What is the probability 

 that at any specified future time he will be within 10 feet of the fence? If it is 

 known that he is not within 10 feet of the south fence, what is the probability 

 that he is not within 10 feet of any of the four fences? Ans. 16/25, 9/20. 



15. Ignoring the refinements, the Rh factor is inherited as described above. 

 The discovery of this factor in 1940 led to an explanation of one type of infant 

 mortality, erythroblastosis. In a large majority of the cases, the father is Rh+, 

 the mother is Rh — , and the child is Rh-|-. Only a fraction of the cases wherein 

 the child is Rh+ and the mother is Rh — , which are potentially erythroblastotic, 

 actually result in trouble; but why some do and others do not is not presently 

 known. Obviously, the father could be either Rh/Rh or Rh/rh, but the mother 

 must be rh/rh. Assume that the population of potential parents is divided for 

 each sex as follows: 



30 per cent RhRh, 60 per cent Rhrh, and 10 per cent rhrh 



What is the expected proportion of potential erythroblastoses among their 

 children? 



3.2 PERMUTATIONS AND COMBINATIONS 



Probability has been calculated in such a way that two numbers 

 need to be determined: (a) the number of single events in the class 

 of events whose probability of occurrence is being determined, and 

 (6) the total number of single events possible under the prescribed 

 conditions or in the mathematical model. For example, the prob- 

 ability of throwing a sum of seven with two unbiased dice is the 

 ratio of the number of single events which give a seven to the total 

 number of ways a sum can be produced. In this instance it is easy 

 to determine those two numbers, but it is not usually easy. The 

 determination of the necessary two numbers often is greatly facili- 

 tated by the use of the mathematical concepts, permutations and 

 combinations. In the process of introducing these concepts, it is con- 

 venient to develop certain useful formulas in terms of abstract let- 

 ters. Thereafter, these symbols will be employed to represent per- 

 sons, heads and tails on a coin, physical objects, etc. 



A set of letters, such as ABC, can differ from another set of the 

 same number of similar marks in one, or both, of two ways: the same 

 letters may appear in a different order, or exactly the same letters 

 may not be present in both sets. For example, ABC and ACB are 

 different orderings of the same three letters, whereas ABC and BCD 



