Binomial Probability Problems

Binomial Probability Problems

A binomial process is a process in which you have only two options each and every time you have to select a member of your sample. For instance, when taking a sample of 5 children out of a large group of children, selecting a child can only be a boy or a girl, there are no other options. Or when you select a sample of 10 light bulbs from a large stock of bulbs and you are interested in knowing whether or not the bulbs in your sample are defective. A bulb can be defective or not defective, there are no other options.

When taking a sample of 10 people in a community and asking them which candidate they supported in the last presidential election, you cannot consider this a binomial process if there were more than two candidates in that election.

Binomial probability problems follow a certain pattern: when selecting 5 children, one question could be: what is the probability that 4 out of the 5 children are boys? Of course we may replace the boys with girls or use numbers other than 4 and 5.

Binomial probability problems can be solved by using the binomial formula [pic]. Recalling the definition of combinations we can reduce the binomial formula to[pic].

In this formula n is the size of the larger group, x the size of the smaller group and p is the probability that an individual child is a boy or an individual light bulb is defective, etc.

Let us look at a few examples of binomial probability problems.

Example 1:

Given that 51.3% of all newly born children are boys, then what is the probability that in a sample of 5 newly born children, exactly 3 are boys?

Solution:

Here we have n = 5, x = 3 and p = .513. Applying the binomial formula we get [pic]

Take the binomial problem of [pic] from example 1

The calculator keystrokes on the TI 30XIIB are:

5

PRB

(

3

x

.513

^

3

x

(

1

-

.513

)

^

2

=

Example 2:

In a large collection of light bulbs we assume that 98% of these bulbs...