You can find the coefficients of an expanded binomial using the numbers in Pascal's triangle. 1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
1 6 15 20 15 6 1
1 7 21 35 35 21 7 1 These are a few rows of Pascals triangle. Now let's look at a few binomials, expanded to the second and third powers. (a+b)2=a2 +2(ab) + b2 notice the coefficients are the numbers in the second row of the triangle above. (a+b)3= a3+3(a2b)+3(ab2)+b3 and once again note that the coefficients are the numberin the third line of Pascal's triangle. The first line, by the way, which is 1,1 is the coefficient of (a+b)1 This will work for any power of the binomial. There are generalized form for non-integer powers.
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Binomial distribution is the basis for the binomial test of statistical significance. It is frequently used to model the number of successes in a sequence of yes or no experiments.
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The symbol for probability of success in a binomial trial is the letter p. It is the symbol used for probability in all statistical testing.
Poisson and Binomial both the distribution are used for defining discrete events.You can tell that Poisson distribution is a subset of Binomial distribution. Binomial is the most preliminary distribution to encounter probability and statistical problems. On the other hand when any event occurs with a fixed time interval and having a fixed average rate then it is Poisson distribution.
It is used when repeated trials are carried out , in which there are only two outcomes (success and failure) and the probability of success is a constant and is independent of the outcomes in other trials.