120 Probability 



tion the cross between the two white-flowered Nemesia plants 

 which produced 86 white and 23 blue-margin plants. 



(x — m)'^ 



x' = 0.884 



In this problem, x^ has a very low value and is certainly not 

 significant. On the basis of the x^ test, this ratio can be con- 

 sidered a true example of a 3 : 1 ratio, and any deviation that 

 it shows from a perfect 3 : 1 ratio can be ascribed purely to 

 chance. 



If the observed ratio happens to be exactly the same as the 

 expected ratio there is no deviation and consequently x" = 0- 

 The greater the deviation, the greater the value of x^? ^^d the 

 smaller the probability that the observed ratio is a true example 

 of the ratio that is being tested. If y^ is 0, the probability is 

 100 per cent that the observed ratio is a true example of the 

 expected ratio; but if y^ is 0.016, the probability is only 90 per 

 cent. That is, if you expect a certain ratio, chance alone will 

 give you an observed ratio with a x^ value of 0.016 in nine times 

 out of ten. If x^ is 0.455, however, the probability of occurrence 

 is only 50 per cent and, if x^ is 1.642, the probability is only 20 

 per cent. Other values of x^ in terms of the probability of occur- 

 rence are given in a table in Chapter 9. 



The term probability of occurrence as used in the x^ test must 

 not be confused with the term probability as used in the bi- 

 nominal expansion. The x^ method states the percentage of 

 cases in which a certain deviation and all greater deviations 

 would occur by chance from a given theoretical ratio for a family 

 consisting of a certain number of individuals. The binominal 

 expansion tells us the probability that we should get a certain 

 number of individuals of each class in a family of a given size if 

 the numbers were segregating into a given ratio. The binominal 

 expansion tells us what would be the chances of getting a family 

 of four boys and two girls if the sexes appear in equal numbers. 

 The x^ method tells us what would be the probability of occur- 



