CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 7. Sampling Distributions and Estimations

TABLE7.3:

Number Who

Correctly Identify

Percolated Coffee

Outcomes

C(correct),

I(incorrect)

Number of Outcomes

0 IIII 1

1 ICII IIIC IICI CIII 4

2 ICCI IICC ICIC CIICCICI CCII 6

3 CICC ICCC CCCI CCIC 4

4 CCCC 1

Using the Multiplication Rule for Independent Events, you know that the probability of getting a certain outcome
when two people guess correctly, like,CICI, is^12 ·^12 ·^12 ·^12 = 161. The table shows six outcomes where two people
guessed correctly so the probability of getting two people who correctly identified the percolated coffee is 166.
Another way to determine the number of ways that exactly two people out of four people can identify the percolated
coffee is simply to count how many ways two people can be selected from four people, or “4 choose 2”:

4 C 2 =

4!

2! 2!

=

24

4

= 6

A graphing calculator can also be used to calculate binomial probabilities.

2 nd[DISTR]↓0:binompdf (This command calculates the binomial probability forksuccesses out ofntrials when
the probability of success on any one trial isp)

A random sample can be treated as a binomial situation if the sample size,n, is small compared to the size of the
population. A rule of thumb to use here is making sure that the sample size is less than 10% of the size of the
population.

A binomial experiment is a probability experiment that satisfies the following conditions:

1. Each trial can have only two outcomes – one known as “success” and the other “failure.”


2. There must be a fixed number,n, of trials.


3. The outcomes of each trial must be independent of each other. The probability of each a “success” doesn’t
   change regardless of what occurred previously.


4. The probability,p, of a success must remain the same for each trial.

The distribution of the random variableXthat counts the number of successes is called a binomial distribution. The
probability that you get exactlyX=ksuccesses is:

P(X=k) =

(

n

k

)

pk( 1 −p)n−k

where

(n

k

)

=k!(nn−!k)!

Let’s return to the coffee experiment and look at the distribution ofX(correct guesses):