Download ppt MAT2323 Pfenning Chap06Sect02

Lecture 14: Chapter 6, Section 2
Finding Probabilities:
More General Rules
General
“And” Rule
More about Conditional Probabilities
Two Types of Error
Independence
©2011 Brooks/Cole, Cengage
Learning
Elementary Statistics: Looking at the Big Picture
1
Looking Back: Review

4 Stages of Statistics



Data Production (discussed in Lectures 1-4)
Displaying and Summarizing (Lectures 5-12)
Probability




Finding Probabilities
Random Variables
Sampling Distributions
Statistical Inference
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L14.2
Probability Rules (Review)
Non-Overlapping “Or” Rule: For any two
non-overlapping events A and B,
P(A or B)=P(A)+P(B).
Independent “And” Rule: For any two
independent events A and B,
P(A and B)=P(A)P(B).
General “Or” Rule: For any two events A and B,
P(A or B)=P(A)+P(B)-P(A and B).
Need “And” Rule that applies even if events are dependent.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L14.3
Example: When Probabilities Can’t Simply
be Multiplied (Review)



Background: In a child’s pocket are 2 quarters and 2
nickels. He randomly picks a coin, does not replace
it, and picks another.
Question: What is the probability of the first and the
second coins both being quarters?
Response: To find the probability of the first and the
second coin being quarters, we can’t multiply 0.5 by
0.5 because after the first coin has been removed, the
probability of the second coin being a quarter is not
0.5: it is 1/3 if the first coin was a quarter, 2/3 if the
first was a nickel.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L14.4
Example: When Probabilities Can’t Simply
be Multiplied
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L14.5
Definition and Notation
Conditional Probability of a second event,
given a first event, is the probability of the
second event occurring, assuming that the
first event has occurred.
P(B given A) denotes the conditional
probability of event B occurring, given
that event A has occurred.
Looking Ahead: Conditional probabilities help us
handle dependent events.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L14.6
Example: Intuiting the General “And” Rule



Background: In a child’s pocket are 2
quarters and 2 nickels. He randomly picks a
coin, does not replace it, and picks another.
Question: What is the probability that the first
and the second coin are quarters?
Response: probability of first a quarter (2/4),
times (conditional) probability that second is a
quarter, given first was a quarter (1/3):
2/4  1/3 = 2/12 = 1/6
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 6.16a p.253
L14.8
Example: Counts Expected if Independent

Background: Students are classified according to gender
and ears pierced or not. A table of expected counts
(
etc.) has been produced.

Question: How different are the observed and expected
counts?
Response: Observed and expected counts are very different
(270 vs. 174, 20 vs. 116, etc.) because ear piercing is very
gender-dependent.

©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 6.18e-j p.254
L14.10
Example: Counts Expected if Independent



Background: Students are classified according to gender
and grade (A or not). A table of expected counts
(
, etc.) has been produced.
Exp
A
not A Total
Obs
A
not A Total
F
15
45
60
F
15
45
60
M
10
30
40
M
10
30
40
Total
25
75
100
Total
25
75
100
Question: How different are the observed and expected
counts?
Response: Counts are identical because gender and grade are
completely independent.
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
Practice: 6.19d-f p.255
L14.12
Lecture Summary
(Finding Probabilities; More General Rules)




General “And” Rule
More about Conditional Probabilities
Two Types of Error
Independence



Testing for independence
Rule for independent events
Counts expected if independent
©2011 Brooks/Cole,
Cengage Learning
Elementary Statistics: Looking at the Big Picture
L14.13