BInomial Theorem

You are probably already familiar with Pascal's Triangle because of the Binomial Theorem. The binomial theorem tells us how to expand powers of a binomial. Expanding powers of a binomial requires repeated application of the distributive property. Each term x and y needs to be multiplied by one of two choices, x or y. This binary choice means that for the square, there are four terms in the binomial expansion, two from x and y being multiplied by x and two from x and y being multiplied by y. If we have a cube, now each of those four terms from the square expansion needs to be multiplied by x and each by y. This gives us a connection to finding combinations.

From the perspective of combinations, we can look at the exponents of each term in the expansion as representing how many factors contribute an x and how many contribute a y to that term and the coefficient tells us how many ways that can be done; that is, which combinations of factors will contribute x and y to the factors. Consider the square, for example: two factors contribute their x terms to generate the x square and this can only be done one way. On the other hand, the xy term is generated by 1 factor for the x and one factor for the y, but this can be done exactly two ways: x from the first factor and y from the second AND x from the second and y from the first. Finally, the y square term can only be generated by pairing the y terms from both factors in the expansion.  

In general, the nth power of the binomial x + y will have in its expansion, terms of degree n where the (n – k)th power of x and kth power of y occurs n choose k many times so that k factors contribute a factor of y and the other n – k factors contribute an x. This is why the binomial theorem tells us we can write the expansion in the following manner:

This theorem can be applied to probability in the specific situation where a sample space consists of the two sole outcomes of a binary event. Taking x and y to represent the probabilities of some binary outcome: x = p the probability that an event occurs and and y = 1 - p is the complement, or probability that event does not occur, we can use the multiplication rule for probability to describe the probability that the event of probability p occurs k times in n observations.

Do you remember when we used the binomial coin experiment calculator for our Coin Flip problem to calculate the probability of having a majority minority in a sample of 10 when the minority makes up only 40% of the population? We added the probabilities from the calculator for getting 6, 7, 8, 9, or 10 from the minority in a sample of 10 and got 16.6%. Now we have a formula to compute the same thing: