As mentioned previously, establishing probabilities where there are only 2 possible outcomes can be done by making use of the binomial expansion:
(p + q)k
Where k is the number draws or iterations. Because p + q = 1, expansion to any value of k also produces a sum equal to 1. Determining the coefficients and exponents for the expansions can be done with relative ease by making use of Pascal's triangle:
The numbers shown above represent the coefficients for values of k ranging from 1 to 8. The exponents can be filled in by iterating all the possible combinations that would sum to k, starting with k + 0, and ending with 0 + k. For example, drawing the coefficients from Pascal's triangle for k=5, and applying the exponents produces the following:
1p5q0 + 5p4q1 + 10p3q2 + 10p2q3 + 5p1q4 + 1p0q5
Which can be simplified to:
p5 + 5p4q + 10p3q2 + 10p2q3 + 5pq4 + q5
What these values represent are the probabilities of all possible outcomes of 5 random and independent draws where each draw produces one of two possible outcomes. Imagine a large bag, full of thousands of marbles (enough marbles that drawing some does not change the proportion of red and blue marbles). One-quarter of these marbles are red (p=0.25), and three-quarters of the marbles are blue (q=0.75). One could use the expansion above, i.e., (p + q)5, to determine the probabilities for each possible combination of 5 marbles that could be drawn:
P{5 red 0 blue} = p5
P{4 red 1 blue} = 5p4q
P{3 red 2 blue} = 10p3q2
P{2 red 3 blue} = 10p2q3
P{1 red 4 blue} = 5pq4
P{0 red 5 blue} = q5
Question 5: Estimate the probability of getting 2 red marbles in a draw of 8 marbles from the bag described above. Estimate the probability of getting any combination of marbles other than 2 red and 6 blue from a draw of 8 marbles from the same bag. What does this probability tell you about sampling error with small sample sizes?
Question 6: Examine the pattern in Pascal's triangle, and explain how you would determine the coefficients for k=9.
Let's move on to cumulative probabilities...
Send comments, suggestions, and corrections to: Derek Zelmer