Transcript 8.1 The Binomial Distribution

Chapter 8
The Binomial &
Geometric Distributions
8.1 The Binomial Distribution
Definition: “The Binomial Setting” :
A situation is said to be a “BINOMIAL SETTING”, if
the following four conditions are met:
1. Each observation is one of TWO possibilities either a success or failure.
2. There is a FIXED number (n) of observations.
3. All observations are INDEPENDENT.
4. The probability of success (p), is the SAME for each
observation.
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8.1 The Binomial Distribution
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Definition: “Binomial Distribution”
The distribution of the count X of successes in the
binomial setting is the BINOMIAL
DISTRIBUTION with parameters n and p.
 n = the number of observations
 p = the probability of success on any one
observation
 A way to symbolically say this: B(n, p)
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8.1 The Binomial Distribution
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Example 8.1: BLOOD TYPES
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Example 8.2: DEALING CARDS
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Example 8.3: INSPECTING SWITCHES
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Example 8.4: AIRCRAFT ENGINE RELIABILITY
8.1 The Binomial Distribution
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Finding Binomial Probabilities
We will use the TI-83/4
 We will use a “by-hand” formula
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Example 8.5: INSPECTING SWITCHES
SRS of 10 switches from a LARGE shipment
 10% of the switches are “bad”
 P(No more than 1 of the 10 switches are “bad”)
 Draw a Probability histogram (on TI-8X)
 Binompdf(n, p, X) and Binomcdf(n, p, X)
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8.1 The Binomial Distribution
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Example 8.6: CORRINE’S FREE THROWS
75% lifetime free-thrower
 12 shots in a key game were takes, and ONLY 7 made
… Is this “unusual”?
 FIST?
 P(X<=7) = ?
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8.1 The Binomial Distribution
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Example 8.7: THREE GIRLS
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Find P(X = 3)
L1 = {0, 1, 2, 3} L2 = binompdf (3, .5, L1)
Plot1…On…Histogram
Xlist:L1 Freq:L2
WINDOW: Xmin: -.5 Xmax: 3.5 Ymin: -.1 Ymax: .4
Xlist:L1 Freq:L3
L3 = binomcdf (3, .5, L1)
WINDOW:
Ymax: 1.1
Graph
8.1 The Binomial Distribution
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Example 8.8: IS CORINNE IN A SLUMP?
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Same 75% free-thrower
Let’s create both the probability distribution and the
cumulative distribution functions.
L1 = {0, 1, 2, … 10, 11, 12}
L2 = binompdf (12, .75, L1) L3 = binomcdf (12, .75, L1)
Xlist:L1 Freq:L2
WINDOW: Xmin: -.5 Xmax: 12.5 Ymin: -.1 Ymax: .3
Xlist:L1 Freq:L3
L3 = binomcdf (3, .5, L1)
WINDOW:
Ymax: 1.1
Graph
8.1 The Binomial Distribution
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Example 8.9: INHERITING BLOOD TYPE
Each child in a family has probability of .25 of having
blood type O.
 P(X = 2)
 FIST?
 List by hand all S-F configuration for 2 S’s in a family of
5.
 Find each probability … multiply by how many ways it
an occur
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8.1 The Binomial Distribution
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The binomial coefficient: An alternative to listing all
10 options from the previous example.
The number of ways of arranging k successes among
n observations is given by:  n 
n!
 
k  k !(n  k )!
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Example:
5
5!
5!
5  4  3  2 1 5  4
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 10
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 2  2!(5  2)! 2!3! (2 1)(3  2 1) 2 1
5
   5 nCr 2  10
 2
8.1 The Binomial Distribution
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The Binomial Probability Formula
If X has the binomial distribution with n
observations and probability p of success on each
observation, the possible values of X are 0, 1, 2, …,
n. If k is any one of these values,
n k
P( X  k )    p (1  p) n k
k 
Example 8.10: DEFECTIVE SWITCHES Part 2
P( X  1)
8.1 The Binomial Distribution
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The Binomial Mean and Standard Deviation
Example 8.11: DEFECTIVE SWITCHES Part 3
8.1 The Binomial Distribution
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The Normal Approximation to the Binomial Distribution – when n is
“large” … Rule of Thumb:
Example 8.12: ATTITUDES TOWARDS SHOPPING
Sample size n = 2500; p = .6 “Agree – I like buying new clothes, but
sopping is often frustrating and time-consuming”
P(X >= 1520)
1 – binomcdf(2500, .6, 1519) … or …
Get mean, standard deviation, and then z, and normalcdf
8.1 The Binomial Distribution
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Simulating Binomial Experiments
 Example
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8.14: CORINNE’S FREE THROWS
p = .75 … n = 12 … P(X <= 7) = 0.1576
randBin(1, .75, 12)
randBin(1, .75, 12)L1:sum(L1)
Simulate 20 games … Compare to .1576
Get class average. Does Law of Large numbers take over?