Probability 111 



also be %. Naturally, p -f g = 1. When (p + g)" is expanded, 

 the coefficients represent the number of cases, the exponent of p 

 represents the number of successes of p, and the exponent of q 

 the number of failures of p or number of successes of q. If three 

 coins are tossed, n is 3, p, the chance of a head, equals %, and q, 

 the chance of not being a head and therefore of being a tail, 

 equals %. The expanded binomial is p^ + Sp^q + ^pq^ + q^- 

 Adding the exponents, we have eight cases in all. One of the 

 eight has all three heads and is represented by the term p^. 

 One of the eight has no heads and is represented by q^. Three 

 of the eight have two heads and one tail, and three have one 

 head and two tails; these two situations are represented by the 

 terms Sp^q and Spq^, respectively. 



In a family of five children (or in five tosses of a coin), what 

 will be the chance that all will be boys (or head, etc.)? Here 

 the binomial becomes {p + q)^, and when this is expanded the 

 result is p^ + dp'^q + lOp^q^ + lOp^q^ + 5pq* + q^. If the 

 coefficients are added, there are thirty-two cases. They will be 

 distributed as follows: 



All boys (or heads) and no girls (or tails) — p^ — 3^'^2 

 Four boys (or heads) and one girl (or tail) — p^g — ^:32 

 Three boys (or heads) and two girls (or tails) — p^q^ — ^%2 

 Two boys (or heads) and three girls (or tails) — p^q^ — 1%2 

 One boy (or head) and four girls (or tails) — pq"^ — %2 

 No boys (or heads) and all girls (or tails) — q^ — J^^2 



In other words, the chance of getting a family of four boys and 

 one girl is 5 out of 32. This can also be arrived at as follows: 



p^ = p • p ' p - p ' p; p = 14) 



•'- P^ = K • 3^ • ^ • H • H = J^2 

 5/g = 5(p ' p ' p ' p ' q); p = ^2 and g = J^; 

 ... 5/g = 5(3^ . K • 3^ • J^ • 3^) = %2 



This same method can also be applied to cases in genetics other 

 than sex. A testcross will illustrate exactly the same situation. 

 If the polled Fi calf is mated mth a horned animal, the theoretical 

 ratio is 1 polled : 1 horned. Therefore, in the binomial, p (poUed) 



