112 Probability 



= 3^ and q (horned or not polled) = 3^. With four offspring the 

 various probabiUties would be determined by (p + q)"^, which 

 equals p^ + 4p^g + 6p^q^ + 4pg^ + 5^- To determine the chance 

 that two animals would be polled and two horned, the term Gp^q^ 

 is used. Substituting the values of p and q gives Q{p • p)iq • q) 

 or 6(3^ • }/i){}/2 ' yQ or %6. Thus, out of everj^ sixteen such 

 families of four, six should be expected which would include two 

 polled and two horned offspring. 



So far, we have considered only 1 : 1 ratios, but the same method 

 can be appUed to 3 : 1 and other ratios. If a homozygous colored 

 tobacco plant is crossed with a white, a 3 : 1 ratio should be ex- 

 pected in the F2. That is, the chance that any one F2 plant would 

 have red flowers is % and the chance that any plant of the F2 

 generation should have white flowers is }i. Therefore, p (the 

 chance of red-flowered plants) = ^i and q (the chance of the fail- 

 ure of a plant to be red-flowered) = }yi. In a family of five plants, 

 the probabilities would be obtained by expanding (p + q)^ and 

 would be: 



p^ = p-p-p-p-p = ^'^i-^-^'^ = 243 fami- 

 lies out of 1024 with 5 red-flowered plants 

 5p^q = 5{p ' p • p ' p)q = 5(M • % • % - Ji)34 = 405 fami- 

 lies out of 1024 with 4 red-flowered and 1 white- 

 flowered plant 

 lOpV ^ io(p • p ■ p)(q • q) = mH • H ' H)iH ' H) = 270 

 families out of 1024 with 3 red-flowered and 2 white- 

 flowered plants 

 lOpV = 10(P • P)(q ■ q- q) = ^H ' H)(H ' M ' M) = 90 

 families out of 1024 with 2 red-flowered and 3 white- 

 flowered plants 

 5p?^ = 5{p)iq • q ■ q- q) = b ■ H{}4 ' M ' K ' Ji) = 15 

 families out of 1024 with 1 red-flowered and 4 white- 

 flowered plants 

 q^ = q-q-q'q-q = M-%'M'M-}4. = ^ family out 

 of 1024 with 5 white-flowered and no red-flowered 

 plants 



In other words, even if a ratio of three red to one white is 

 expected, a family of five white-flowered plants is not impossible 

 for it would be expected to happen in one case out of 1024. 

 Other ratios than the 1 : 1 testcross ratio and the 3 : 1 mono- 



