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

POPULATION

SAMPLE

Probability

Inferential Statistics

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

• Calculate the following probabilities?

– 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

• Calculate the following probabilities?

– 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

Visually

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


2. Draw normal curve, approximately locate Z

score, shade in the area for which you are

finding the percentage.


z=+1.56

Approx. 92% of cases below 125 IQ


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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