Announcements
Finite Probability
Wednesday, November 2nd
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MyMathLab 7 is due tonight!
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Problem Set 7 is due Friday Nov 4
Today: Sec. 7.3: Binomial Trials
Identify experiments with binomial probability
distributions
Use the binomial formula to solve binomial word problems
Next Class: Sec. 7.3: Binomial Trials II
Cherveny
Nov 2
Math 1004: Probability
Bernoulli Trials
Definition
A Bernoulli Trial is one trial of a probability experiment that has
only two outcomes, commonly called “success” and “failure” with
a fixed probability “p” of success and “q = 1 − p” of failure each
trial.
Example
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Roll a die and observe either “5” or “not 5”.
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Test a newly made light bulb from the assembly line, observe
either “good” or “defective”.
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Draw three cards from a deck of cards, observe either “two
kings and a queen” or “something else”.
Cherveny
Nov 2
Math 1004: Probability
Last Class: Flipping a Coin 7 Times
Example
We flipped a fair coin 7 times and observed X = total number of
heads flipped. We found the probability distribution
7
1
P(X = k) = C (7, k)
2
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Why is every flip a Bernoulli trial?
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What are p and q?
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Expand (p + q)7 by the binomial theorem from section 5.7 (!).
Answer:
(p + q)7 = p 7 + C (7, 1)p 6 q + C (7, 2)p 5 q 2 + · · · + C (7, 6)pq 6 + q 7
Cherveny
Nov 2
Math 1004: Probability
Binomial Theorem
Theorem
If X is the random variable counting the number of successes in n
Bernoulli trials of an experiment with probability of success is p
and failure is q, then
P(X = k) = C (n, k)p k q n−k
Note: We just did this with the coin for n = 7, p = q = 21 .
Cherveny
Nov 2
Math 1004: Probability
Rolling Dice
Example
Suppose you roll two fair dice 14 times.
(a) What is the probability of getting a pair of 6’s exactly 5 times?
(b) What is the probability of getting a sum of 4 exactly three
times?
(c) What is the probability of getting a doubles at least 12 times?
Answer:
(a) C (14, 5)
(b) C (14, 2)
(c) C (14, 12)
Cherveny
1 5
36
3 3
36
1 12
6
35 9
36
33 11
36
5 2
+
6
C (14, 13)
Nov 2
1 13
6
5 1
6
+
Math 1004: Probability
1 14
6
Practice
1. You draw a card from a 52 card deck, look at it, and put it
back (shuffled).
(a) What is the probability of getting 3 hearts when you do this 8
times?
(b) What is the probability of getting 2 kings when you do this 10
times?
2. Fourteen percent of U.S. residents are in their twenties. You
choose 6 U.S. residents at random. What is the probability
that you have at least two people in their twenties?
3. You have an urn with 3 red chips and 6 white chips. Pick
three chips from the urn. Then pick three chips again. Then
pick three chips again.
(a) What is the probability exactly two chips are white on the first
pick?
(b) What is the probability you get exactly two white chips all
three picks if you don’t replace chips? Is this Binomial?
(c) What is the probability you get exactly two white chips three
times out of five picks if you do replace chips? Is this Binomial?
Cherveny
Nov 2
Math 1004: Probability
Practice Answers
1. You draw a card from a 52 card deck, look at it, and put it
back (shuffled).
(a) C (8, 3)
(b) C (10, 2)
1 3 3 5
4
4
1 2 12 8
13
13
2. 1 − (.86)6 − C (6, 1)(.14)(.86)5
3. You have an urn with 3 red chips and 6 white chips. Pick
three chips from the urn. Then pick three chips again. Then
pick three chips again.
(a)
C (6,2)·C (3,1)
C (9,3)
(b) Not binomial;
C (6,2)·C (3,1)
C (9,3)
(c) Binomial; C (5, 3)
Cherveny
C (4,2)·C (2,1) C (2,2)·C (1,1)
·
C (6,3)
C (3,3)
3 2
C (6,2)·C (3,1)
C (6,2)·C (3,1)
1
−
C (9,3)
C (9,3)
Nov 2
·
Math 1004: Probability