Computing Binomial Probabilities

In Example 1, the number of correct guesses may be 0, 1, 2, 3, 4 or 5. How likely can a guesser get all 5 questions right? The answer is .001, or about once in every 1000 tries. How about the likelihood of getting 2 out of 5 questions right? The answer is .264, approximately a quarter of the time. The following probability distribution table  gives the likelihood or probability of each possible value of X.

$$\begin{array}{c\vert c\vert c\vert c\vert c\vert c} P(X=0) ... ... \\ \hline .237 & .395 & .264 & .088 & .015 & .001 \end{array}$$ (4.1)

You can compute these probabilities yourself by successively substituting j=0, 1, 2, 3, 4 and 5 in the formula

$$P(X=j)= \frac{5!}{j!\; (5-j)!} (1/4)^j\;(3/4)^{4-j},$$ (4.2)

where $j!=(j)(j-1)(j-2)\cdots (1)$ and noting that 0!=1. This formula is called the binomial probability distribution function  (or pdf) for n=5 and p=1/4. To compute the probabilities for Examples 2 and 3, you will need the binomial pdf for general n  and p:  

$$P(X=j)= \frac{n!}{j!\; (n-j)!} p^j\;(1-p)^{n-j}, \; j=0, 1, 2,\ldots, n.$$ (4.3)

Example 3 (Con't): 10 TV sets were sold in one day. What is the probability that 3 extended warranties were sold? Using Equation  4.3 with n=10, p=.20, and j=3, we get

$$P(X=3) = \frac{10!}{3!\;7!} (.20)^3\;(.80)^7 = .201$$

Exercise: Five percent of videos rented at Campus Video are returned late. If 30 videos were rented during the last hour, what is the probability that
a. 2 will be returned late
b. none will be returned late

Binomial probabilities may also be presented in a cumulative form $P(X\leq j) = P(X=0)+P(X=1)+\cdots +P(X=j)$. Here is the cumulative distribution function (cdf)   for the binomial distribution with n=5 and p=1/4.

$$\begin{array}{c\vert c\vert c\vert c\vert c\vert c} P(X\leq... ... \\ \hline .237 & .632 & .896 & .984 & .999 & 1.00 \end{array}$$ (4.4)

Do you see how to compute ( 4.4) from ( 4.1)? The TI-83 calculator can compute either the binomial pdf or the binomial cdf (see Sections  4.4.3 and  4.4.4 at the end of this chapter).

Exercise: Five percent of videos rented at Campus Video are returned late. If 30 videos were rented during the last hour, what is the probability that
a. 2 or fewer will be returned late
b. 5 or more will be returned late