introduction to probability

Introduction to Probability
STA 250

Building to Inferential Statistics
• Our goal is to use samples to answer questions
about populations
• This is called inferential statistics
• Built around the concept of probability
• Specifically, the relationship between samples
and populations are usually defined in terms of
probability
If you know the makeup of a population you can
determine the probability of obtaining a specific
sample

Basics
• The Relationship Visually
P
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Inferential Statistics

Probability

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What is Probability
• Probability
– Expected relative frequency of a particular
outcome

• Outcome
– The result of an experiment

Steps to Finding the Probability of an
Event
1. Determine number of possible successful
outcomes
2. Determine number of all possible outcomes
3. Divide number of possible successful
outcomes (Step 1) by number of all possible
outcomes (Step 2)

Rules of Probability
Rule 1: Probabilities range between 0 and 1
– That is when you add the probabilities of success
and failure they should equal 1

Example
• What is probability of getting number 3 or
lower on a throw of a die?
1.Determine number of possible successful
outcomes (1, 2, 3)=3
2.Determine number of all possible outcomes
(1, 2, 3, 4, 5, and 6)=6
3.Divide Successful Outcomes/All Outcomes: 3/6=.5

Example
• Calculate the following probabilities
– Getting heads with a single coin flip P(h)
– Rolling a 2 with a single die P(2)
– Pulling a heart from a deck of cards P(heart)
– Rolling a 7 with two dice P(7)

Example
• Calculate the following probabilities?
– Getting heads with a single coin flip P(h)
P(h) = 1/2
– Rolling a 2 with a single die P(2)
P(2) = 1/6
– Pulling a heart from a deck of cards P(heart)
P(heart) = 13/52
– Rolling a 7 with two dice P(7)
P(7) = 7/36

Rule 2: The Addition Rule
• The probability of alternate events is equal to
the sum of the probabilities of the individual
events
P(A or B) = P(A) + P(B)
• Rolling a 3 or a 4 with one die
• P(3 or 4) = 1/6 + 1/6 = 2/6 = .333

Example
• What is the probability of drawing an ace of
spades (A) or an ace of hearts (B) from a deck
of cards?
– P(A or B) = P(A) + P(B)
– P(A) = 1/52 = 0.02
– P(B) = 1/52 = 0.02

– P(A or B) = 0.02 + 0.02 = 0.04 (or 4%)

Example
– Rolling a 2 or a 5 or higher with a single die P(2 or
5 or higher)
– Pulling a heart or a spade from a deck of cards
P(heart or spade)

Example
– Rolling a 2 or a 5 or higher with a single die P(2 or
5 or higher)
P(2 or 5 or higher) = 1/6 + 2/6 = 3/6
– Pulling a heart or a spade from a deck of cards
P(heart or spade)
P(heart or spade) = 13/52 + 13/52 = 26/52

Rule 3: Addition Rule for Joint Occurrences
• The probability of events that can happen at
the same time is equal to the sum of the
individual probabilities minus the joint
probability
P(A or B) = P(A) + P(B) – P(A and B)
• Drawing a king and a heart for a deck of cards
P(K or H) = 4/52 + 13/52 – 1/52 = 16/52 = .307

Rule 4: Multiplication Rule Compound Events
• The probability of a event given another event
has occurred equals the product of the
individual probabilities
P(A then B) = P(A) * P(B)
• Getting a heads then a tails with a single coin
• P(H then T) = 1/2 * 1/2 = 1/4 = .25

Example
• What is the probability of drawing an ace of
spades (A) and an ace of hearts (B) from a
deck of cards?
– P(A and B) = P(A) x P(B)
– P(A) = 1/52 = 0.02
– P(B) = 1/52 = 0.02

– P(A and B) = 0.02 x 0.02 = 0.0004 (or 0.04%)

• Rule 5: Multiplication with replacement
Same as Rule 4 but you account for changes in
probability caused by first event

Lets Gamble
• The probability of winning the millions on a
three reel slot machine
0.000300763

• Blackjack (card game with best odds)

Remember
• A frequency distribution represents an entire

population
• Different parts of the graph refer to different parts of
the population
• Proportions and probability are equivalent
• Because of this a particular portion of the graph refers

to a particular probability in the population

Example
• If we had a population (N=10) with the
following scores 1, 1, 2, 3, 3, 4, 4, 4, 5, 6 and
wanted to take a sample of n=1 what is the
probability of obtaining a score greater than 4
p(X>4)?
• How many possible successful outcomes
• How many possible outcomes

The Same Applies to the Normal Curve
• We can identify specific locations in a normal
distribution using z-scores
• Using z-scores, the properties of normal
distributions, and the unit normal table we
can find what proportions of scores fall
between specific scores in a distribution
• Because proportions and probability are
equivalent we can determine what the
probability of falling in that area is

The Unit Normal Table
• Column A
– Z-score

• Column B
– Proportion in body
(larger part of
distribution

• Column C
– Proportion in Tail
(smaller part of
distribution

• Column D
– Proportion between
mean and z-score

FINDING PROBABILITY UNDER THE NORMAL
CURVE

1. Convert raw score into Z score (if
necessary):
• Example:
– For IQ: μ=100, σ=16
– If a person has an IQ of 125, what
percentage of people have a lower IQ?

– Z=(125 – 100)/16 = +1.56

CURVE
2. Draw normal curve, approximately locate Z
score, shade in the area for which you are
finding the percentage.

CURVE

Approx. 92% of cases
below 125 IQ



z=+1.56

CURVE

4. Find exact percentage using
unit normal table.

.9406

Find the Following Percentages
•
•
•
•

Higher than 2.00
Higher than -2.00
Lower than 2.49
Between -1.0 and 1.0

Find the Following Percentages
•
•
•
•

Higher than 2.00 = .0228 (Column C)
Higher than -2.00 = .9772 (Column B)
Lower than 2.49 = .9936 (Column B)
Between -1.0 and 1.0 = .6826 (Column D)

introduction to probability

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