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Commentary on Probability:
Ever gamble or go to the casinos? What is the probability that you will win given a lotto ticket?
Welcome to one of the authors of statistics.
As confounding as probability may seem in the beginning, it is a major underlying factor of
inferential statistics (remember your definition of inferential statistics!). Some examples of how
probability plays a role in our daily lives: How long will these winter ‘blues” last? When is the
best time to try & drive back home from work? How many times do I have to tell my kids to
clean their room before they will actually do it? What is the likelihood of getting an “A” in class?
I could go on & on. All of these are probabilities. When you hear about the likelihood that
someone will develop a mental illness, the percentage given is originally based on a probability.
For all of the examples above (& then some), you can re-phrase them as probability statements:
What is the probability of someone developing heart disease? What is the probability of a baby
being born w/ birth defects? What is the probability of getting into a certain school? Etc… For
most of these examples, too, it is not just one factor that determines probability, but multiple
factors. For heart disease, family history, lifestyle, eating habits, work environment; you have to
consider each factor as an individual probability & then combine these probabilities together to
come up w/ a single factor.
As mentioned above on the important role probability plays in inferential statistics, we are
evaluating sample data to evaluate the probability of the same or similar results applying to the
full population. For example, we can use a sample & evaluate the probability of a trial drug
resulting in certain side effects. The drug hits the general market, & the probability of side
effects are then applied to the population as a whole.
There are a couple of tenets on probability that you want to focus on:
1. Each possible outcome must have the same chance of being chosen. Changing the chance
of being chosen changes the probability (see below). The most appropriate & statistical
means of sampling is called Random Sampling: Requires that each individual in the
population has an equal chance of being selected. A second requirement, necessary for
many statistical formulas, states that the probabilities must stay constant from one
selection to the next if more than one individual is selected. The vast majority of research
utilizes random sampling as a part of their research method.
2. There is sampling with replacement & sampling without replacement. For instance,
given a deck of cards you have 52 cards. You have a 4 out of 52 (4/52) chance of pulling
out a Queen. Statistically, you must replace the card that you pulled from the deck to
have the same chance (4/52) of pulling out a King. That is sampling with replacement. If
you pull out a Queen (4/52) & then keep the Queen out of the deck, your chances of
pulling out a King next changes (4/51) since you are no longer working with a full deck.
That is sampling without replacement. Unless a problem indicates that the sampling is
done without replacement, then you should revert to sampling with replacement (or
random sampling) for any problem.
3. There are two simple rules that apply to calculating probability problems.
a) The Add-Or rule. If you are asked to calculate one probability or another, then you
use addition.
b) If you are asked to calculate the probability of one event and another event occurring,
then you would multiply those probabilities.
Don’t forget to double check if you are working with sampling with replacement or
sampling without replacement. If the problem does not specify sampling without
replacement, then you can assume that you will be sampling with replacement.
4. Steps to assessing a probability statement:
a) What am I trying to obtain? Example: What is the probability of obtaining a king
when drawing from a full deck?
The written probability statement would look like this: p(King)
b) What am I drawing from? The full deck of playing cards – 52
c) Am I drawing with or without replacement? Since we are working with only one
probability statement, then this does not come into play.
d) Develop your fraction: the number of chances of obtaining your king from the full
deck: 4/52. Why? Because there are 4 kings out of a full deck of 52.
e) Convert the fraction into a decimal: 4/52 = 0.076923
When you are working with probability, you will want to go as far to the right of
the decimal point as possible to obtain the most accurate conclusion!!
f) Convert the decimal into a percentage by moving the decimal point 2 places to the
right: 7.6923%
Therefore, the probability of drawing a King from a full deck of cards is
7.6923%
Simple probability: So as you review your materials & text on probability, you will
first run into simple examples, such as the probability of pulling one yellow marble
out of a jar containing 20 yellow marbles & 40 black marbles. This is a simple, one
dimensional probability problem. You will be pulling only one marble out, but since
there are 20 yellow marbles in the jar of 60 marbles altogether, then you have a 20 out
of 60 chance of obtaining a yellow marble. The next step would be to convert your
fraction into a decimal by dividing 20/60 = .3333 & then convert to a percentage by
moving the decimal over 2 spaces to the right = 33.33%. So you have a 33.33%
chance of pulling out a yellow marble.
5. Steps when working with more than one probability:
a) What am I trying to obtain? Example: (bringing in the and/or rule into play). For
example: What is the probability of obtaining a king and a queen?
b) When working with a probability statement involving more than one
probability, you want to calculate one probability at a time. Do not combine
probability fractions in the beginning. The written probability statement would
look like this: p(King) x p(Queen). Why are we multiplying? We are observing the
and/or rule. If the question involves an and, then you multiply. If the question
involves or, then you add your probabilities together.
c) What am I drawing from? The full deck of playing cards – 52
d) Am I drawing with or without replacement? Good question. If it does not specify,
then review any additional information to see if it tells you whether or not you are
drawing with or without replacement. If you see the phrase “random selection” or if
the statement does not say specifically without replacement, then you will be drawing
with replacement.
REMEMBER!! In random selection, each subject must have the same
probability of being selected. If you are sampling w/out replacement, then it
changes the probability for the next draw!!
e) Develop your fraction: When working with replacement, what is the probability of
drawing a King or a Queen from a deck of cards? Take each probability at a time. The
number of chances of obtaining a King from the full deck: 4/52. Why? Because there
are 4 Kings out of a full deck of 52. For your second probability statement, your
fraction will still be 4/52. Why? Because we replaced the first King we drew.
f) Convert the fractions individually into decimals: p(King) = 0.076923 + (we add
here since the statement said a King or a Queen p(Queen) = 0.076923
When you are working with probability, you will want to go as far to the right of
the decimal point as possible to obtain the most accurate conclusion!!
g) Complete your calculations using the and/or rule: 0.076923 + 0.076923 = .153846
h) Convert the decimal into a percentage by moving the decimal point 2 places to the
right: 15.3846%
Complex probability: Problems that involve multiple factors: Example: What is the
probability of pulling a King, Queen & Jack from a deck of cards in consecutive
draws without replacement? This probability problem involves multiple factors for
which you have to obtain each probability individually & then join together these
individual probabilities to obtain a single probability value associated w/ these
multiple factors. So in this example, we know that a deck of cards has 52 cards in
them. There are 4 Kings, 4 Queens & 4 Jacks. I look at drawing my King first, which
would be 4 out of 52 (4/52 = .076923). Now before I continue the same process w/
the Queen & Jack, I see that the problem stated without replacement. This means that
I lay that King aside, leaving me now w/ only 51 cards. So the next probability of
obtaining a Queen would be 4 out of 51 (4/51 = .0784313). I repeat then again w/ my
Jack, knowing that I now have 2 cards sitting out. 4/50 = .08. Since I am dealing w/
multiple probabilities, I need to look & see whether I add or subtract based upon the
and/or rule. If it says “or,” I add. If it says “and,” I multiply. In this case, it says
“and,” so that means I will multiply my probabilities together (NOTE: Do not convert
your individual probabilities into percentages until after you have applied the
“and/or” rule to the problem & come up w/ a final value)
( .076923) x (.0784313) x (.08) = .0004826 (then convert to percentage) = .04826%
meaning that there is a .04826% chance that I will draw a King, Queen & a Jack in
consecutive draws without replacement.