776 CHAPTER 38. PROBABILITY

is given by

P(X = k) =

(nk

)pkqn−k

then X is said to have a binomial distribution.

Note that the sumn

∑k=0

P(X = k)

needs to equal 1 because the random variable must achieve one of the numbers 0,1,2, · · · ,n.This occurs by the binomial theorem,

n

∑k=0

P(X = k) =n

∑k=0

(nk

)pkqn−k = (p+q)n = 1n = 1.

There is a general principle of counting which should be mentioned. Suppose you havem “positions” and n different things. How many ways are there to fill the m positions withthe n things? There are n choices for the first, and having filled this position, there aren− 1 left to place in the second. Thus the number of ways to fill the first two positionsis n(n−1) . Then, having filled these two, there are now n− 2 things left to place in thethird position and so there are n(n−1)(n−2) ways to fill the first three of these positions.Continue this way till you run out of positions to fill. How many ways of filling them doyou obtain? You see that there are n(n−1)(n−2) · · ·(n−m+1) ways to do it. This iscalled permutations of n things taken m at a time. See the exercise below.

Example 38.5.2 In a class of 12 students who are arranged in three rows of four students,what is the probability that the particular four students, Eliphaz, Elihu, Zophar, and Bildadwill occupy the front four seats?

There are 4! ways for them to occupy these four seats in some order. There are 12 ·11 ·10 ·9 ways to fill these seats in some order. Therefore, the probability is

4!12(11)(10)(9)

=1

495

Of course, you don’t care about order in this problem so you could also do this in terms ofcombinations of n things taken m at a time.

1( 12!4!8!

) = 1495

There is exactly one way to select these four students for the first four seats and then thereare 12!/(4!8!) ways to fill these seats.

Example 38.5.3 In the above example involving 12 students, it is absolutely necessary fordisciplinary reasons that Eliphaz must not sit next to Elihu. If the students file in and sitdown randomly, what is the probability that Eliphaz ends up on the front right seat whenviewed by the teacher and is not sitting next to Elihu?