Transcript Sullivan Chapter 6 - Whitehall District Schools

Chapter 6
Section 2
The Binomial
Probability Distribution
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 1 of 31
Chapter 6 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 2 of 31
Chapter 6 – Section 2
A binomial experiment has the following 4 rules:
1. Only 2 possible results (success/failure, yes/no,
on/off, etc)
2. Each trial is independent of the others
3. Fixed number of trials
4. Probability of success remains constant with each
trial
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 3 of 31
Chapter 6 – Section 2
Example
 A card is drawn from a deck. A “success” is for that
card to be a heart … a “failure” is for any other suit
 The card is then put back into the deck
 A second card is drawn from the deck with the same
definition of success.
 The second card is put back into the deck
 We continue for 10 cards
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 4 of 31
Chapter 6 – Section 2
● Notation used for binomial distributions
 The number of trials is represented by n
 The probability of a success is represented by p
 The total number of successes in n trials is
represented by X
● Because there cannot be a negative number of
successes, and because there cannot be more
than n successes (out of n attempts)
0≤X≤n
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 5 of 31
Chapter 6 – Section 2
● The word “success” does not mean that this is a
good outcome or that we want this to be the
outcome: it’s just what we’re looking for
● A “success” in our card drawing experiment is to
draw a heart
● If we are counting hearts, then this is the
outcome that we are measuring
● There is no good or bad meaning to “success”
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 6 of 31
Chapter 6 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 7 of 31
Chapter 6 – Section 2
● The general formula for the binomial
probabilities is just this
● For P(x), the probability of x successes, the
probability is
 The number of ways of choosing x out of n, times
 The probability of x successes, times
 The probability of n-x failures
● This formula is
P(x) = nCx px (1 – p)n-x
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 8 of 31
Chapter 6 – Section 2
Example
● A student guesses at random on a multiple
choice quiz
 There are n = 10 questions in total
 There are 5 choices per question so that the
probability of success p = 1/5 = .2
● What is the probability that the student gets 6
questions correct?
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 9 of 31
Chapter 6 – Section 2
Example continued
● This is a binomial experiment
 There are a finite number n = 10 of trials
 Each trial has two outcomes (a correct guess and an
incorrect guess)
 The probability of success is independent from trial to
trial (every one is a random guess)
 The probability of success p = .2 is the same for each
trial
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 10 of 31
Chapter 6 – Section 2
Example continued
● The probability of 6 correct guesses is
P(x) = nCx px (1 – p)n-x
= 6C10 .26 .84
= 210 • .000064 • .4096
= .005505
● This is less than a 1% chance
● In fact, the chance of getting 6 or more correct
(i.e. a passing score) is also less than 1%
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 11 of 31
Chapter 6 – Section 2
● Binomial calculations can be difficult because of
the large numbers (the nCx) times the small
numbers (the px and (1-p)n-x)
● It is possible to use tables to look up these
probabilities
● It is best to use a calculator routine or a software
program to compute these probabilities
●Stop
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 12 of 31
Chapter 6 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 13 of 31
Chapter 6 – Section 2
● The mean of a binomial distribution
μX = n p
Example
 There are 10 questions
 The probability of success is .20 on each one
 Then the expected number of successes would be
10 • .20 = 2
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 14 of 31
Chapter 6 – Section 2
● The standard deviation and variance of a
binomial distribution
● The standard deviation:
σX = √ n p (1 – p)
and the variance is
σX2 = n p (1 – p)
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 15 of 31
Chapter 6 – Section 2
● For our random guessing on a quiz problem
 n = 10
 p = .2
 x=6
● Therefore
 The mean is np = 10 • .2 = 2
 The variance is np(1-p) = 10 • .2 • .8 = .16
 The standard deviation is √.16 = .4
● Remember the empirical rule? A passing grade of
6 is 10 standard deviations from the mean …how
do we interpret this?
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 16 of 31
Chapter 6 – Section 2
● Learning objectives
1

Determine whether a probability experiment is a
binomial experiment
2 Compute probabilities of binomial experiments
3
 Compute the mean and standard deviation of a
binomial random variable
4
 Construct binomial probability histograms
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 17 of 31
Chapter 6 – Section 2
● With the formula for the binomial probabilities
P(x), we can construct histograms for the
binomial distribution
● There are three different shapes for these
histograms
 When p < .5, the histogram is skewed right
 When p = .5, the histogram is symmetric
 When p > .5, the histogram is skewed left
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 18 of 31
Chapter 6 – Section 2
● For n = 10 and p = .2 (skewed right)
 Mean = 2
 Standard deviation = .4
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 19 of 31
Chapter 6 – Section 2
● For n = 10 and p = .5 (symmetric)
 Mean = 5
 Standard deviation = .5
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 20 of 31
Chapter 6 – Section 2
● For n = 10 and p = .8 (skewed left)
 Mean = 8
 Standard deviation = .4
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 21 of 31
Chapter 6 – Section 2
● Despite binomial distributions being skewed, the
histograms appear more and more bell shaped
as n gets larger
● This will be important!
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 22 of 31
Quiz question
● A doctor tries an experimental drug on 24
cancer patients to see how effective it is. If each
patient has a 30% probability of defeating
cancer, what is the probability of at most 22 of
the 24 patients defeating the cancer?
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 23 of 31
Summary: Chapter 6 – Section 2
● Binomial random variables model a series of
independent trials, each of which can be a
success or a failure, each of which has the same
probability of success
● The binomial random variable has mean equal
to np and variance equal to np(1-p)
Sullivan – Statistics: Informed Decisions Using Data – 2nd Edition – Chapter 6 Section 2 – Slide 24 of 31