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Probability and
Counting Rules
4-1: Sample Spaces and Probability
What is Probability?
• Probability – the chance of an event occurring.
• This can be represented as a fraction, decimal or a percent.
• Probability is used in games of chance such as dice, cards, slot
machines, lotteries, etc.
• Probability also applies in fields of insurance, investments and
weather forecasting.
• Predictions about many things are made based on the
probability of an event occurring.
• Probability is the foundation of inferential statistics.
Sample Space
• Probability Experiment – a chance process that
leads to well-defined results called “outcomes”.
• Outcome – is a result of a single trial of a
probability experiment.
• Sample Space – the set of all possible outcomes
of a probability experiment.
Finding the Sample Space
Finding the Sample Space
Finding the Sample Space
• Find the sample space for a probability experiment where a
coin is flipped four times.
• Use “H” for heads and “T” for tails.
• Another way to do this is by using a “tree diagram”.
Finding the Sample Space
• Another way to find
the sample space is
by using a tree
diagram.
• Tree Diagram – a
device consisting of
line segments
emanating from the
starting point and
also from the
outcome point. It is
used to determine
all possible
outcomes of a
probability
experiment.
Finding the Sample Space
• Using a tree diagram, find the sample space for a probability
experiment where a coin is flipped four times.
• Use “H” for heads and “T” for tails.
Events
• An event consists of a set of outcomes of a probability
experiment.
• An event may include just one outcome or more than one
outcome.
• Simple Event – an event with just one outcome
• Compound Event – an event with more than one outcome
• An example: Consider rolling a single die…
• Describe the event of rolling a 6.
• Describe the event of rolling an odd number.
Interpretations of Probability
1.
Classical Probability – uses sample spaces to determine the
numerical probability that an event will happen.
• You do not actually have to perform the experiment to determine
that probability.
• This is the focus of this section.
• Assumes that all outcomes are equally likely.
2.
Empirical Probability – relies on actual experience to determine
the likelihood of outcomes. (Also called relative frequency
probability.)
• Example: Roll a die 6,000 times and record the actual frequency that
each outcome occurs. This determines the probability of each
outcome.
3.
Subjective Probability – is based on an educated guess or
estimate, employing opinions an inexact information.
• Example: A doctor estimates that there is a 30% chance that you will
need an operation.
Classical Probability
• Assumes that all outcomes in the sample space
are equally likely to occur.
• Equally Likely Events – events that have the
same probability of occurring.
Probability Values
• Probability can be expressed as a fraction,
decimal or percent.
• Probability as a fraction or decimal is a value
between and including 0 and 1.
• If P(E) is the probability of an event occurring,
0 ≤ P(E) ≤ 1
• If an event E cannot occur, its probability is 0: P(E) = 0.
• If an event E is certain, its probability is 1: P(E) = 1.
Examples of Calculating Probability
Example of Probability
• If you flip a coin, there are two possible outcomes, “heads”
and “tails”. The probability that you get heads can be
expressed three different ways:
• “one-half” (fraction)
• 0.5 (decimal)
• 50% (percent)
Rounding Rule for Probability
•
•
•
•
Option – Write as a fraction in simplest form.
Option – Write as a decimal to THREE decimal places.
Option – Write as a percent to ONE decimal place.
If the probability is a very small decimal such as 0.0000587,
round to the first non-zero digit after the decimal point, such
as 0.00006.
• If you convert this to a percent, simply move the decimal two
places to the right: 0.006%
Examples of Calculating Probability
Example of Calculating Probability
“and” versus “or”
• “and” means “at the same time”
• Example: If you were asked the probability of finding a queen and a
heart from a deck of cards, you are looking for the probability of
finding the queen of hearts.
• This is 1 possibility.
• Inclusive “or” means one or the other but they could both occur at
the same time
• Example: If you were asked the probability of finding a queen or a
heart from a deck of cards, you are looking for all queens, all hearts
and the queen of hearts.
• This is 4 + 13 – 1 = 16 possibilities.
• Exclusive “or” means one or the other and it is not possible for them
to occur at the same time
• Example: If you were asked the probability of finding a queen or a
king from a deck of cards. You cannot get both with a single card.
• This is 4 + 4 = 8 possibilities.
Sum of Probabilities
• Rule: The sum of the probabilities of all the
outcomes in a sample space is 1.
• Consider the probability experiment of rolling a
single die.
• The sample space is {1, 2, 3, 4, 5, 6}
• The probability of each outcome is 1/6.
Complementary Events
• Consider the probability experiment of rolling a
single die. The sample space consists of the
outcomes {1, 2, 3, 4, 5, 6}.
• If we define the event E as getting an odd-numbered
outcome, then E = {1, 3, 5}
• Then the complement of E is not getting an odd
number or instead getting an even number.
• E-bar mean “not E”.
Example of Complementary Events
Rule for Complementary Events
• Rule: The sum of the probability of event E and
the probability of the complement of event E is
always one.
• Sometimes it is easier to find the probability of
the complement of E and subtract it from one
than to find the probability of E, itself.
Example of Using the Probability of
Complements
Empirical Probability
• Empirical Probability – relies on actual experience to
determine the likelihood of outcomes. (Also called
relative frequency probability.)
• In empirical probability, a researcher might actually roll a
die 6,000 times and observe the various frequencies and
use these frequencies to determine probability of an
outcome.
• Suppose a researcher for AAA asked 50 people who plan
to travel over Thanksgiving and how they plan to get to
their destination. The results of this survey is shown
below and would be used to compute probability.
Formula for Empirical Probability
Example of Empirical Probability
Example of Empirical Probability
Empirical Probability and Relative
Frequency
• Empirical probabilities can also be found by
using a relative frequency distribution as shown
below in our Thanksgiving travel example.
Law of Large Numbers
• If we conduct an empirical probability experiment where we
flip a coin, we would expect to get heads “half” of the time. If
we complete 50 flips, we would expect to get heads 25 times.
However, that will not necessarily occur because of laws of
chance.
• The law of large numbers says that the more times we
perform this experiment the closer we will be to getting heads
half of the time if the coin is fair.
• In reality, casinos use this phenomenon. If they notice that a
particular pair of dice does not seem to get each outcome an
equal number of times as classical probability suggests, they
will remove the dice as they are not considered “fair”.
Subjective Probability
• Subjective Probability – is based on an educated guess or
estimate, employing opinions an inexact information.
• This is based on a person’s expertise.
• A doctor may estimate the probability that surgery will be needed
based on his experience.
• An accountant may estimate the probability that a client would
be audited based on his expertise.
• A plumber may estimate the probability that your pipes will
spring a leak if you do not fix them.
• Subjective Probability is imprecise and only an educated
guess.
Probability and Mainstream Media
• An area in which people fail to understand probability is risk taking.
• Mainstream media use shock and outrage to appeal to our emotions
about certain issues. These emotions may make listeners feel like a
certain issue has a significant probability of occurring and hurting
you.
• Be aware of true statistical probabilities calculated by reputable
resources.
• Example: News highlights rising crime. In reality there has been no
change in the probability that you will be affected by crime in the
last 20 years.
• Example: News highlights the risk of pesticides. In reality, the
probability of dying by pesticide poisoning is 1 in 200,000. The risk
of dying from a heart attack is 1 in 3, but it is not covered in the
media to nearly the same extent.