Probability 113 



hybrid ratio are frequently encountered in genetics. The method 

 would be the same for them. 



It will be noted that the sum of the probabilities of all possible 

 events gives a total probability of 1. Thus, in the family of 5 

 children in which a ratio of 1 boy to 1 girl is involved, the 

 sum of all the possible probabilities (i.e., the probability of 5 

 boys + the probability of 4 boys and 1 girl, etc.) is ^%2, or 1, 

 whereas in a 3 : 1 ratio with 5 plants, the sum of the probabilities 

 is ^^-^1024? 01' 1- The value of 1 is certainty, and unless an 

 event is certain, its probability is expressed as a fraction or 

 decimal. The expression p -\- q = 1 must be true, for 1 — p = q, 

 which means that the certainty of an event less the probability 

 that it will happen equals the probability that it will not happen. 

 If an event is certain, the probability that it will happen is 1 and 

 the probability that it will not happen is 0, so 1 — = 1. 



In Nicotiana Sanderae a testcross showed 672 colored and no 

 white plants. If the colored parent were heterozygous, would it 

 be possible for so many colored plants and no white plants to be 

 produced? If it were possible, what then would be the prob- 

 ability? Since this is a testcross, the ratio would be 1 : 1; there- 

 fore p = y2 and q = y2- The binomial (p H- q)^"^^ could be 

 expanded, but this would not be necessary, since the first term 

 would give all the information needed. The probability of get- 

 ting a family of 672 colored plants from a testcross would be 

 p^'^ or (%)^'^-. It is very obvious that the probability of getting 

 such a result from a cross between a heterozygote and a recessive 

 is so small that one would be well justified in assuming that the 

 colored plant tested was homozygous. 



This last example shows that although the method of binomial 

 expansion is theoretically correct it has some very practical limi- 

 tations. In practice, it is not usual to apply this method to popu- 

 lations over 50, as it is far too cumbersome. In fact, even with 

 numbers between 15 and 50, the binominal expansion becomes 

 unwieldy. Warwick, however, has worked out the probabilities 

 for numbers up to 50, and by using his tables, much labor is 

 avoided. For example, in Nemesia strumosa, orange flower color 

 is dominant over white. A cross of an orange by a white gave a 

 ratio of 25 : 17, when the expected testcross ratio would have 

 been 21 : 21. An examination of Warwick's table shows that 

 the probability of getting a ratio of 25 dominants and 17 re- 



