Section 6.2 ~
Basics of Probability
Introduction to Probability and Statistics
Ms. Young
Sec. 6.2
Objective

After this section you will know how to find
probabilities using theoretical and relative frequency
methods and understand how to construct basic
probability distributions.
Sec. 6.2
Basics of Probability

Outcomes – the most basic possible results of observations or
experiments


Ex. ~ There are four outcomes of tossing two coins, HH, HT, TH,
or TT
Event – a collection of one or more outcomes that share a
property of interest

Ex. ~ Suppose you are only interested in the number of heads that
appear when you toss two coins. There would be three events, 0
heads (TT), 1 head (HT or TH), or two heads (HH)
Sec. 6.2
Basics of Probability Cont’d…

To express a probability, you use numbers
between 0 and 1 inclusive

A probability of 0 would represent an event that is
impossible


A probability of 1 would represent an event that is
certain to occur


Ex. ~ Death and taxes!
The probability of an event is written as P(event)


Ex. ~ Meeting a married bachelor
Ex. ~ The probability of landing a head on a coin toss
would be written as P(H) = .5
The scale to the right shows common expressions
used to represent probabilities based on their level
in comparison to 0 and 1



Ex. ~ A probability of .95 indicates that an event is
very likely to occur (95 out of 100 times)
Ex. ~ A probability of .30 indicates that an event is
unlikely to occur (30 out of 100 times)
Ex. ~ A probability of .01 describes an event that is
very unlikely to occur (1 out of 100 times)
Sec. 6.2
Theoretical Probabilities


Theoretical probabilities are probabilities that deal with
equally likely outcomes (i.e., tossing a fair coin, rolling a fair die,
spinning a roulette wheel, etc.)
Calculating theoretical probabilities:
Step 1: Count the total number of possible outcomes
 Step 2: Among all the possible outcomes, count the number of
ways the event of interest, A, can occur
 Step 3: Determine the probability, P(A), from

P( A) 
number of ways A can occur
total number of outcomes
Sec. 6.2
Example 1

Suppose you select a person at random from a large group.
What is the probability that the person has a birthday in July?
Assume that there are 365 days in a year.

Since all birthdays are equally likely, we can use the 3 step process
for calculating theoretical probabilities:



Step 1: Each possible birthday represents an outcome, so there are
365 possible outcomes
Step 2: July has 31 days, so 31 of the 365 possible outcomes represent
the event of a July birthday
Step 3: The probability that a randomly selected person has a birthday
in July is
P( A) 
number of ways A can occur
total number of outcomes
P(July birthday) 
31
 0.0849
365
Sec. 6.2
Theoretical Probabilities Cont’d…

Counting Outcomes – the total number of outcomes can be found by
raising the individual outcome to the number of processes

Ex. ~ What is the total number of outcomes of tossing two coins?


Each coin has 2 outcomes (H or T), and there are 2 coins (2 tosses or
processes), so there are 4 possible outcomes
(22 = 4) when tossing two coins
Ex. ~ What is the total number of outcomes of tossing three coins?

Each coin has 2 outcomes (H or T), and there are 3 coins (3 tosses or
processes), so there are 8 possible outcomes (23 = 8) when tossing three
coins
Sec. 6.2
Example 2
How many outcomes are there if you roll a fair die and toss a fair coin?
The first process, rolling a fair die, has six possible outcomes, 1, 2, 3, 4, 5, or
6, and the second process, tossing a fair coin, has two possible outcomes
(H or T).
The total number of outcomes would be 12 ( 61 × 21).
What is the probability of rolling two 1’s (snake eyes) when two fair dice
are rolled?
Rolling a single die has 6 equally likely outcomes, so rolling two dice
has a total of 36 outcomes (62 = 36).
Of the 36 outcomes, the event of interest (two 1’s) can only occur
one way, so the probability of rolling two 1’s is
P(two 1's) 
1
 0.0278
36
Sec. 6.2
Example 3
What is the probability that in a randomly selected family with three children,
the oldest child is a boy, the second child is a girl, and the youngest child is a
girl? Assume that having boys and girls is equally likely.
There are two possible outcomes for each birth: boy or girl
For a family with three children, there would be 8 possible outcomes
(23 = 8)
BBB, BBG, BGG, GBB, GBG, GGB, GGG
The probability of the birth order being BGG is
P(BGG) 
1
 0.125
8
Sec. 6.2
Relative Frequency Probabilities

Another method to determine probabilities is to approximate
the probability of an event, A, occurring. This is known as the
relative frequency (or empirical) method.


Ex. ~ If we observe that it rains an average of 100 days per year,
we can estimate the probability of it raining on a randomly selected
day to be approximately .274 (100/365)
Here is the general rule for this method:
Step 1: Repeat or observe a process many times and count the
number of times the event of interest, A, occurs.
 Step 2: Estimate P(A) by

number of times A occurred
P( A) 
total number of observations
Sec. 6.2
Example 4
Geological records indicate that a river has crested above a particular
high flood level four times in the past 2,000 years. What is the relative
frequency probability that the river will crest above the high flood
level next year?
P(river will flood next year) 
4
1

 0.002
2000 500
Because a flood of this magnitude occurs on average once every 500 years,
it is called a “500-year flood.”
The probability of having a flood of this magnitude in any given year is
1/500, or 0.002.
Sec. 6.2
Subjective Probabilities &
Summary of Different Methods of Finding Probabilities

A third method for determining probabilities is to estimate a
subjective probability using experience or intuition


Ex. ~ You could make a subjective estimate of the probability that a friend
will be married in the next year or the probability that getting a good grade
in statistics will help you get the job that you want
Three approaches to Finding Probability

Theoretical probability – when all outcomes are equally likely, divide the
number of ways an event can occur by the total number of outcomes
P( A) 
number of ways A can occur
total number of outcomes

Relative frequency probability – based on observations or experiments.
Divide the number of times the event occurred by the total number of
observations
number of times A occurred
P( A) 
total number of observations

Subjective probability – estimating based on experience or intuition
Sec. 6.2
Example 5
Identify the method that resulted in the following statements.
a. The chance that you will get married in the next year is zero.
Subjective because it’s based on a feeling
b. Based on government data, the chance of dying in an automobile accident
is 1 in 7,000 (per year).
Relative frequency probability because it’s based on observations
on passed automobile accidents
c. The chance of rolling a 7 with a twelve-sided die is 1/12.
Theoretical probability because it is based on assuming that a
fair twelve sided die is equally likely to land on any of its twelve
sides.
Sec. 6.2
Probability of an Event Not Occurring

Sometimes you might be interested in finding the probability
that a particular event or outcome does not occur
 Ex. ~ The probability of a wrong answer on a multiple choice
question with five possible answers


The probability of answering it correctly would be .2 (1/5), so
the probability of not answering it correctly would be .8 (4/5)
The complement of an event, A, expressed as A, consists of all
outcomes in which A does not occur.
 The sum of the probabilities of A and A must be 1, so the probability of A
can be given by
P( A)  1  P( A)
Sec. 6.2
Example 6
In a grocery store the scanning system was successful 384
out of 419 times. What is the probability that the scanner
will not work?
384
P ( A)  1 
 0.084
419
Sec. 6.2
Probability Distributions

A probability distribution is a visual display of the probabilities of certain
events occurring in the form of a table or a histogram

Ex. ~ Suppose you toss two coins simultaneously. Because each coin can land one of
two ways (H or T), there are 4 possible outcomes (HH, TT, HT, & TH). The following
table represents the outcomes and probabilities:

Out of the 4 outcomes, there are 3 events, 2 heads (HH), 1 head (HT or TH), and 0
heads (TT). These probabilities result in a probability distribution which can
represented as a table or a histogram:
Sec. 6.2
Probability Distributions

Steps to making a probability distribution:
Step 1: List all possible outcomes
 Step 2: Identify outcomes that represent the same event. Find the
probability of each event.
 Step 3: Make a table or a histogram in which one column (or x-axis)
represents the events and the other column (or y-axis) represents the
probability


Example 7:

Make a probability distribution table for the number of heads that occur
when three coins are tossed simultaneously.

Step 1: List all possible outcomes


Since there are 3 coins, there are a total of 8 outcomes (23 = 8): HHH, HHT, HTH, HTT,
THH, THT, TTH, and TTT (refer to p.241 to see how these outcomes were constructed)
Step 2: Identify outcomes that represent the same event. Find the probability
of each.

Since we are interested in the number of heads that occur, there would be 4 events, 0
heads (1/8 = .125), 1 head (3/8 = .375), 2 heads (3/8 = .375), or 3 heads (1/8 = .125)
Sec. 6.2
Probability Distributions

Example 7 Cont’d:

Make a probability distribution table for the number of heads that occur
when three coins are tossed simultaneously.

Step 3: Make a table