Essay on Law of Probability and Binomial Expression!
The law of the probability of coincident independent events states that the chance or probability of the simultaneous occurrence of two or more independent events equal to the product of the probability that each will occur separately. Thus, when two independent events occur with the probability p and q respectively then the probability of their joint occurrence is pq.
For example, if 10 coins are tossed together the most frequent results will be 5 heads and 5 tails, 4 heads and 6 tails, 3 heads and 7 tails, etc., occur less often and cither 10 heads or 10 tails is rare. If we draw a graph of the results of the heads and tails in the tossess of coin then we would have a symmetrical graph (curve). This symmetrical or normal probability curve is usually derived by plotting on expansion of expression (a+b)n or (p+q)n, which is known as binomial expression. Here, a or p stands for the probability of the head, while b or q stands for the probability of the tail and n stands for number of coins. According to the different values of n, following expansion of (p+q)n can be received :
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(p+q)1 = p+q
(p+q)2 = p2+2pq+q2
(p+q)3 = p3+3p2q+3pq2+q3
(p+q)4 = p4+4p3q+6p2q2+4pq3+q4
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(p+q)5 = p5+5p4q+10p3q2+10p2q3+5pq4+q5.
Here, the coefficients (the numbers before the powers of p and q) express the number of ways a particular outcome is obtained. An easy method for evaluating the coefficients of the expanded terms of the binomial expression is through the use of Pascal’s triangle, which is shown below:
Thus, Pascal’s triangle provides the coefficients of the binomial expression—that is, the number of possible outcomes of various combinations of events.
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Binomial expression has its wide use in population genetics and ecology. For example, suppose we have a population of insects, exactly 40% of which are infected with a given virus X. If we take samples of k = 5 insects each and examine each insect separately for presence of virus, what distribution of samples could we expect if the probability of infection of each insect in a sample were independent from that of other insects in the sample?
In this case p=0.4, the proportion infected, and q= 0.6, the proportion not infected. It is assumed that the population is large and that the question of whether sampling is with or without replacement is irrelevant for practical purposes. The expected proportions would be the expansion of the binomial:
(p + q)k = (0.4+0.6)5
With the aid of Pascal’s triangle this, expansion is the probability space
{p5+5p4q+10p3q2+10p2q3 + 5pq5+q5}
Or (0.4)5 + 5 (0.4)4 (0.6) + 10 (0.4)3 (0.6)2 +10 (0 4)2 (0 6)3 + 5 (0.4) (0.6)4 + (0.6)5
representing the expected proportion of samples of five infected insects, four infected and one non-infected insects, three infected and two non-infected insects, and so on. Thus, in “five-infected insects” there is a probability of occurrence—in this case (0.4)5 = 0.01024.