http://www.ck12.org Chapter 3. An Introduction to Probability
3.6 Basic Counting Rules
Learning Objectives
- Understand the definition of random sampling.
- Calculate ordered arrangements using factorials.
- Calculate combinations and permutations.
- Calculate probabilities with factorials.
Inferential Statisticsis a method of statistics that consists of drawing conclusions about a population based on
information obtained from a subset or sample of the population. The main reason a sample of the population is only
taken rather than the entire population (a census) is because it is less costly and it can be done more quickly than a
census. In addition, because of the inability to actually reach everyone in a census, a sample can actually be more
accurate than a census.
Once a statistician decides that a sampling is appropriate, the next step is to decide how to select the sample. That
is, what procedure should we use to select the sample from the population? The most important characteristic of any
sample is that it must be a very good representation of the population. It would not make sense to use the average
height of basketball players to make an inference about the average height of the entire US population. It would not
be also reasonable to estimate the average income of the entire state of California by sampling the average income
of the wealthy residents of Beverly Hills. Therefore, the goal of sampling is to obtain a representative sample. For
now, we will only study one powerful way of taking a sample from a population. It is calledrandom sampling.
Random Sampling
A random sampling is a procedure in which each sample of a given size is equally likely to be the one selected. Any
sample that is obtained by random sampling is called arandom sample.
In other words, ifnelements are selected from a population in such a way that every set ofnelements in the
population has an equal probability of being selected, then thenelements form a random sample.
Example:
Suppose you randomly select 4 cards from an ordinary deck of 52 cards and all the cards selected are kings. Would
you conclude that the deck is still an ordinary deck or do you conclude that the deck is not an ordinary one and
probably contains more than 4 kings?
Solution:
The answer depends on how the cards were drawn. It is possible that the 4 kings were intentionally put on top of the
deck and hence drawing 4 kings is not unusual, it is actually certain. However, if the deck was shuffled well, getting
4 kings is highly improbable. The point of this example is that if you want to select a random sample of 4 cards to
draw an inference about a population, the 52 cards deck, it is important that you knowhowthe sample was selected
from the deck.
Example: