Transcript The Binomial Probability Distribution and Related Topics

Review
• 3.3 Percentiles and Box-and-Whisker Plots
• Chapter 4 Elementary Probability Theory
• Chapter 5 The Binomial Probability Distribution
and Related Topics
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|1
3.3 Percentiles and Box-andWhisker Plots
• Percentiles
• Quartiles
• Box-and-Whisker Plots
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|2
Probability Assignment
• Assignment by intuition – based on intuition,
experience, or judgment.
• Assignment by relative frequency –
P(A) = Relative Frequency = f
• Assignment for equally likely outcomes
n
Num berof Outcom esFavorableto Event A
P( A) 
Total Num berof Outcom es
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|3
The Sum Rule and The Complement Rule
• The sum of the probabilities of all the simple
events in the sample space must equal 1.
• The complement of event A is the event that A
does not occur, denoted by Ac
• P(Ac) = 1 – P(A)
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|4
• Multiplication Rule for Independent Events
P( A and B)  P( A)  P( B)
• General Multiplication Rule – For all events
(independent or not):
P( A and B)  P( A)  P( B | A)
P( A and B)  P( B)  P( A | B)
• Conditional Probability (when
P( B)  0
P( A and B)
P( A | B) 
P( B)
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
):
5|5
Mutually Exclusive Events
• Two events are mutually exclusive if they
cannot occur at the same time.
• Mutually Exclusive = Disjoint
• If A and B are mutually exclusive, then
P(A and B) = 0
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|6
Addition Rules
• If A and B are mutually exclusive, then P(A or
B) = P(A) + P(B).
• If A and B are not mutually exclusive, then P(A or
B) = P(A) + P(B) – P(A and B).
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|7
Multiplication Rule for Counting
This rule extends to outcomes involving three, four, or more series of events.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|8
Factorials
• For counting numbers 1, 2, 3, …
• ! is read “factorial”
– So for example, 5! is read “five factorial”
• n! = n * (n-1) * (n-2) * … * 3 * 2 * 1
– So for example, 5! = 5 * 4 * 3 * 2 * 1 = 120
• 1! = 1
• 0! = 1
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5|9
Permutations
• Permutation: ordered grouping of objects.
• Counting Rule for Permutations
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 10
Combinations
• A combination is a grouping that pays no
attention to order.
• Counting Rule for Combinations
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 11
Chapter 5 The Binomial Probability
Distribution and Related Topics
• Introduction to Random Variables and
Probability Distribution
• Binomial Probabilities
• Additional Properties of the Binomial Distribution
• The Geometric and Poisson Probability
Distributions
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 12
5.1 Introduction to Random Variables
and Probability Distribution
• Statistical Experiments – any process by which
measurements are obtained.
• A quantitative variable, x, is a random variable if
its value is determined by the outcome of a
random experiment.
• Random variables can be discrete or
continuous.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 13
Random Variables and
Their Probability Distributions
• Discrete random variables – can take on only
a countable or finite number of values.
• Continuous random variables – can take on
countless values in an interval on the real
line
• Probability distributions of random variables
– An assignment of probabilities to the
specific values or a range of values for a
random variable.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 14
Means and Standard Deviations for
Discrete Probability Distributions
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 15
Finding µ and σ for
Linear Functions of x
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 16
Combining Random Variables
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 17
5.2 Binomial Experiments
1) There are a fixed number of trials. This is
denoted by n.
2) The n trials are independent and repeated
under identical conditions.
3) Each trial has two outcomes:
S = success
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
F = failure
5 | 18
Binomial Experiments
4) For each trial, the probability of success, p,
remains the same. Thus, the probability of
failure is 1 – p = q.
5) The central problem is to determine the
probability of r successes out of n trials.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 19
Binomial Probability Formula
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 20
Determining Binomial Probabilities
1) Use the Binomial Probability Formula.
2) Use Table 3 of Appendix II.
3) Use technology.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 21
5.3 Graphing a Binomial
Distribution
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 22
Mean and Standard Deviation of a
Binomial Distribution
  np
  npq
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 23
5.4 The Geometric Distribution
• Suppose that rather than repeat a fixed number
of trials, we repeat the experiment until the first
success.
• Examples:
– Flip a coin until we observe the first head
– Roll a die until we observe the first 5
– Randomly select DVDs off a production line
until we find the first defective disk
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 24
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 25
The Poisson Distribution
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 26
Poisson Approximation to the Binomial
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
5 | 27