After reading this article you will learn about the calculation of probability using binomial distribution.
Binomial expansion is of great help in solving genetical problems related to probability. If there are 2 events with alternate independent events having probabilities p and q, then in n number of trials, the probabilities of various combinations of events is given by (p + q)n where p + q = 1
For illustration, we may write
(i) (p + q)2 = p2 + 2pq + q2
(ii) (p + q)3 = p3 + 3 p2 q + 3 pq2 + q3
(iii) (p + q)4 = p4 + 4 p3 q + 6 p2 q2 + 4pq3 + q4
(iv) (p + q)5 = p5 + 5 p4 q + 10 p3 q3 + 10 p2 q + 5pq4 + q5
It follows from the above equations that:
(i) The number of terms in an expanded binomial is n+1.
(ii) The coefficient of the first term is 1.
(iii) The power of the first factor (p) is n and that of (q) is zero.
(iv) The (r+1)th term may be written as:
For example, 3rd term in (iii) equation in the earlier case can be derived as:
Coefficient = nCr = 4C2 = 4 x 3/2 x 1 = 6
Index = p2q2 and 3rd term = 6 p2q2
Similarly, other terms can be derived. The practical application of this formula can be demonstrated by expanding
p = frequency of boys = 1/2
q = frequency of girls = 1/2
This binomial expansion shows the probability of various combinations of boys and girls in a family of 4 disregarding the sequence of children. For example, 6/16 p2q2 tells that the probability of having 2 boys and 2 girls is 6/16 in a family of 4 children.
The 6 tells the number of arrangements in which 2 boys and 2 girls are possible as : BBGG, BGBG, BGGB, GGBB, GBGB and GBBG. The probability of a particular combination may be obtained from a formula based on the appropriate term in the binomial expansion of (p+q)n;
For example, the probability of obtaining 2 tall and 2 dwarf plants in a typical monogenic F2 population where the probability of tall plants, p = 3/4 and that of dwarf plants, q = 1/4, will be as given below:
