Transcript document

Probability
Important Terms
Probability experiment:
An action through which counts, measurements
or responses are obtained
Sample space:
The set of all possible outcomes
Event:
A subset of the sample space.
Outcome:
The result of a single trial
Example
Probability experiment:
Sample space:
Roll a die
{1 2 3 4 5 6}
Event: { Die is even } = { 2 4 6 }
Outcome:
{4}
Practice
• Use a tree diagram to develop the sample
space that results from rolling two sixsided dice.
Tree Diagrams
Two dice are rolled.
Describe the
sample space.
1
1st roll
1 2 3 4 5 6
2
1 2 3 4 5 6
Start
3
4
12 3 4 5 6 1 2 3 4 5 6
5
6
1 2 3 4 5 6
1 2 3 4 5 6
2nd roll
36 outcomes
Sample Space and Probabilities
Two dice are rolled and the sum is noted.
1,1
1,2
1,3
1,4
1,5
1,6
2,1
2,2
2,3
2,4
2,5
2,6
3,1
3,2
3,3
3,4
3,5
3,6
4,1
4,2
4,3
4,4
4,5
4,6
5,1
5,2
5,3
5,4
5,5
5,6
6,1
6,2
6,3
6,4
6,5
6,6
Find the probability the sum is 4.
3/36 = 1/12 = 0.083
Find the probability the sum is 11.
2/36 = 1/18 = 0.056
Find the probability the sum is 4 or 11.
5/36 = 0.139
Ice Cream Sundaes
Consider an ice cream shop with 31
flavors of ice cream and 15 different
toppings.
How many different sundaes can you
make if you use 1 scoop of ice cream
and two different toppings?
What if you use 3 different scoops and
2 toppings?
More multiplication principle
How many different outcomes can you
have if you flip a coin 3 times?
How many different outcomes can you
have if you flip a coin and roll a die?
Theoretical
P(A) =
number if ways A can occur
total number of outcomes
In a bag you have 3 red marbles, 2 blue
marbles and 7 yellow marbles. If you
select one marble at random,
P(red) = 3 / (3+2+7) = 3/12 = 1/4
Theoretical Examples
•Using a standard deck of cards, find the
probability of the following.
•Selecting a seven
•Selecting a diamond
•Selecting a diamond, heart, club or spade
•Selecting a face card
Empirical
Probability is based on observations or
experiments.
Empirical examples
• A pond contains three types of fish: bluegills,
redgills and crappies. You catch 40 fish and
record the type. The following frequency
distribution shows your results.
Fish Type Number of times caught
Bluegill
13
Redgill
17
Crappy
10
If you catch a fish, what is the probability that it is
a redgill?
Empirical examples
What is the probability of getting a
bluegill?
What is the probability of not getting a
crappy?
Subjective
• Subjective probability results from
educated guesses, intuition and estimates.
• A doctor’s prediction that a patient has a 90%
chance of full recovery
• A business analyst predicting an employee strike
being 0.25
Summary
• Classical (Theoretical)
• The number of outcomes in a sample space is
known and each outcome is equally likely to
occur.
• Empirical (Statistical)
• The frequency of outcomes in the sample space is
estimated from experimentation.
• Subjective (Intuition)
• Probabilities result from intuition, educated
guesses, and estimates.
Probability
• If P(E) = 0, then event E is impossible.
• If P(E) = 1, then event E is certain.
0  P(E)  1
Impossible
0
Even
.5
Certain
1
Complementary Events
The complement of event E is event E´.
E´ consists of all the events in the sample space that
are not in event E.
E
E´
P(E´) = 1 - P(E)
Example
The day’s production consists of 12 cars, 5 of
which are defective. If one car is selected at
random, find the probability it is not defective.
Solution:
P(defective) = 5/12
P(not defective) = 1 - 5/12 = 7/12 = 0.583
Examples
What is the probability that a family
with 3 children does not have 2 boys
and 1 girl?
What is the probability that you do not
get a pair of sixes when you roll 2 dice?
Probability Distributions
A discrete probability distribution lists each possible
value of the random variable, together with its probability.
A survey asks a sample
of families how many
vehicles each owns. number of
vehicles
x
0
1
2
3
P(x)
0.004
0.435
0.355
0.206
Properties of a probability distribution
• Each probability must be between 0 and 1, inclusive.
• The sum of all probabilities is 1.
Example
A company tracks the number of sales new
employees make each day during a 100-day
probationary period. The results for one new
employee are shown below. Construct and graph
the probability distribution.
Sales per day, x
0 1 2 3 4 5 6 7
number of days, f 16 19 15 21 9 10 8 2
Example
Decide whether each distribution is a
probability distribution. Explain your
reasoning.
x
5
6
7
P(x) 1/16 5/8 ¼
8
3/16
Example
Decide whether each distribution is a
probability distribution. Explain your
reasoning.
x
1
2
3
4
P(x) 0.09 0.36 0.49 0.06
Odds
Odds for an event A are
P(A)
P(not A)
In sports we often look at wins over
losses.
Independent Events
• Two events are independent if the
occurrence (or non-occurrence) of one of
the events does not affect the probability
of the occurrence of the other event.
• Events that are not independent are
dependent.
Examples
Independent Events
A = Being female
B = Having type O blood
A = 1st child is a boy
B = 2nd child is a boy
Dependent Events
A = taking an aspirin each day
B = having a heart attack
A = being a female
B = being under 64” tall
Examples
Determine if the following events are independent or
dependent.
1. 12 cars are on a production line where 5 are defective
and 2 cars are selected at random.
A = first car is defective
B = second car is defective.
2.
Two dice are rolled.
A = first is a 4 and B = second is a 4
Multiplication Rule
To find the probability that two events, A and B will occur
in sequence, multiply the probability A occurs by the
conditional probability B occurs, given A has occurred.
P(A and B) = P(A) x P(B given A)
Two cars are selected from a production line of 12 where 5
are defective. Find the probability both cars are defective.
A = first car is defective B = second car is defective.
P(A) = 5/12
P(B given A) = 4/11
P(A and B) = 5/12 x 4/11 = 5/33 = 0.1515
Multiplication Rule
Two dice are rolled. Find the probability both are 4’s.
A = first die is a 4 and B = second die is a 4.
Independent Events
• When two events A and B are independent,
then P (A and B) = P(A) x P(B)
***Note for independent events P(B) and P(B
given A) are the same.
Compare “A and B” to “A or B”
The compound event “A and B” means that A and
B both occur in the same trial. Use the
multiplication rule to find P(A and B).
The compound event “A or B” means either A
can occur without B, B can occur without A or
both A and B can occur. Use the addition rule to
find P(A or B).
B
A
A and B
B
A
A or B
Mutually Exclusive Events
Two events, A and B, are mutually exclusive if
they cannot occur in the same trial.
A = A person is under 21 years old
B = A person is running for the U.S. Senate
A = A person was born in Philadelphia
B = A person was born in Houston
A
B
Mutually exclusive
P(A and B) = 0
When event A occurs it excludes event B in the same trial.
Non-Mutually Exclusive Events
If two events can occur in the same trial, they are
non-mutually exclusive.
A = A person is under 25 years old
B = A person is a lawyer
A = A person was born in Philadelphia
B = A person watches West Wing on TV
A and B
Non-mutually exclusive
P(A and B) ≠ 0
A
B
Examples
Determine whether the events are mutually
exclusive or not.
1. Roll a die
A: Roll a 3 B: Roll a 4
2. Select a student
A: select a male student B: select a nursing
major
3. Select a blood donor
A: donor is type O
B: donor is female
Examples
4. Select a card from a standard deck
A: the card is a jack
B: the card is a face card
5. Select a student
A: the student is 16 years old
B: the student has blue eyes
6. Select a registered vehicle
A: the vehicle is a Ford
B: the vehicle is a Toyota
The Addition Rule
The probability that one or the other of two events
will occur is:
P(A) + P(B) – P(A and B)
A card is drawn from a deck. Find the
probability it is a king or it is red.
A = the card is a king
B = the card is red.
P(A) = 4/52 and P(B) = 26/52
but P(A and B) = 2/52
P(A or B) = 4/52 + 26/52 – 2/52
= 28/52 = 0.538
The Addition Rule
A card is drawn from a deck. Find the
probability the card is a king or a 10.
A = the card is a king B = the card is a 10.
P(A) = 4/52 and P(B) = 4/52 and P(A and B) = 0/52
P(A or B) = 4/52 + 4/52 – 0/52 = 8/52 = 0.054
When events are mutually exclusive,
P(A or B) = P(A) + P(B)
Examples
1. A die is rolled. Find the probability of
rolling a 6 or an odd number.
-are the events mutually exclusive?
-find P(A), P(B) and, if necessary,
P(A and B)
-use the addition rule to find the
probability
Example
2. A card is selected from a standard deck.
Find the probability that the card is a face
card or a a heart.
-are the events mutually exclusive?
-find P(A), P(B) and, if necessary,
P(A and B)
-use the addition rule to find the probability
Contingency Table
The results of responses when a sample of adults in
3 cities was asked if they liked a new juice is:
Omaha
Yes
100
No
125
Undecided 75
Total
300
Seattle
150
130
170
450
Miami
150
95
5
250
Total
400
350
250
1000
One of the responses is selected at random. Find:
1. P(Miami and Yes)
3. P(Miami or Yes)
2. P(Miami and Seattle)
4. P(Miami or Seattle)
Contingency Table
Omaha
Yes
100
No
125
Undecided 75
Total
300
Seattle
150
130
170
450
Miami
150
95
5
250
Total
400
350
250
1000
One of the responses is selected at random. Find:
1. P(Miami and Yes)
= 250/1000 • 150/250 = 150/1000
= 0.15
2. P(Miami and Seattle) = 0
Contingency Table
Omaha
Yes
100
No
125
Undecided 75
Total
300
Seattle
150
130
170
450
Miami
150
95
5
250
Total
400
350
250
1000
3 P(Miami or Yes)
250/1000 + 400/1000 – 150/1000
= 500/1000 = 0.5
4. P(Miami or
Seattle)
250/1000 + 450/1000 – 0/1000
= 700/1000 = 0.7
Summary
For complementary events P(E') = 1 - P(E)
Subtract the probability of the event from one.
The probability both of two events occur
P(A and B) = P(A) • P(B given A)
Multiply the probability of the first event by the
conditional probability the second event occurs,
given the first occurred.
Summary
Probability at least one of two events occur
P(A or B) = P(A) + P(B) - P(A and B)
Add the simple probabilities, but to prevent double
counting, don’t forget to subtract the probability of
both occurring.
Law of Large Numbers
• As an experiment is repeated over and
over, the empirical probability of an event
approaches the theoretical (actual)
probability of the event.
Expected Value
Average- what you would expect given
a probability distribution.
To find the expected value, multiply
each event value by its probability and
add
Example
X
0
1
2
3
P(X)
.25
.50
.15
.10
0(.25) + 1(.5) + 2(.15) + 3(.10) = 1.1
Example
A bus arrives at a bus stop at noon, 12:20 and
1:00. You arrive at the bus stop at random
times between noon and 1:00 every day so
all arrival times are equally likely.
a. What is the probability that you will arrive at
the bus stop between noon and 12:20?
What is the mean wait time in that case?
b. What is the probability that you will
arrive at the bus stop between 12:20
and 1:00?
What is your mean wait time in that
case?
c. Overall, what is your expected waiting
time for the bus?
d. Would you expect your waiting time to
be longer or shorter if the bus arrived
at equally spaced intervals (say noon,
12:30 and 1:00)?
Example
You are given 10 to 1 odds against
tossing three heads in three tosses of a
fair coin, meaning you win $10 if you
succeed and you lose $1 if you fail.
Find the expected value (to you) of the
game.
Would you expect to lose or win money
on 1 game? in 100 games?
The probability of tossing 3 heads in 3 tosses is
1/8, and the probability of not tossing 3
heads is 7/8.
The expected value of the game is
(10 x 1/8) + (-1 x 7/8) = $0.375
-expect to win around 38 cents per game on
average
The outcome of one game cannot be predicted,
over 100 games you should expect to win.
Example
You are given 10 to 1 odds against rolling a
double number (for example two 1’s or two
2’s) with the roll of two dice, meaning you
win $10 is you succeed and you lose $1 if you
fail.
Find the expected value (to you) of the game.
Would you expect to lose or win money on 1
game? in 100 games?
Terms
The gambler’s fallacy is the mistaken
belief that a streak of bad luck makes a
person “due” for a streak of good luck.
The house edge is the amount the casino,
or house, can expect to earn per dollar
bet.
Example
Suppose you toss a fair coin 100 times,
getting 42 heads and 58 tails, which is
16 more tails then heads.
a. Explain why, on your next toss, the
difference between the number of
heads and tails is as likely to grow to 17
as it is to shrink to 15.
b.
c.
Extend your explanation from part a to
explain why, if you toss a coin 1000 more
times, the final difference is as likely to be
larger than 16 as it is to be smaller than 16.
Suppose you are betting on heads with
each coin toss. After the first 100 tosses,
you are well on the losing side. Explain
why, if you continue to bet, you will most
likely remain on the losing side.
How is this answer related to the gambler’s
fallacy?
Example
In a large casino, the house wins on its
blackjack tables with a probability of 50.7%.
All bets at blackjack are 1 to 1: If you win,
you win the amount you bet, if you lose,
you lose the amount you bet.
a. What is the expected value to you of a
single game?
What is the house edge?
b.
If you played 100 games of blackjack in an
evening, betting $1 on each hand, how
much would you expect to win or lose?
c.
If you played 100 games of blackjack in an
evening, betting $5 on each hand, how
much would you expect to win or lose?
d. If patrons bet $1,000,000 on blackjack
in one evening, how much should the
casino expect to earn?
7E
Counting and Probability
Factorial
3! = 3 x 2 x 1 = 6
5! = 5 x 4 x 3 x 2 x 1= 120
X! = x(x-1)(x-2)…1
3! Is read as “3 factorial”
On the TI-83 the factorial key is located by
pressing
MATH arrow over to PROB down to 4:!
Arrangements
When we put things in order it is important to
know if repetition is allowed.
How many 3 digit codes can be made using the
numbers 0 – 9 if repetition is allowed?
If repetition is not allowed?
Examples
How many license plates can be made using 3
digits and 3 letters, repetition is allowed?
How many license plates can be made using 3
digits and 3 letters if the numbers cannot
repeat?
How many 7 character passwords can be made
using letters and numbers, repetition is
allowed?
Permutations
A permutation is an ordered arrangement.
The number of permutations for n objects is n!
n! = n (n – 1) (n – 2)…..3 • 2 • 1
The number of permutations of n
objects taken r at a time (r  n) is:
Example
You are required to read 5 books from a list of 8.
In how many different orders can you do so?
There are 6720 permutations of 8 books reading 5.
Combinations
A combination is a selection of r objects from a
group of n objects.
The number of combinations
of n objects taken r at a time is
Example
You are required to read 5 books from a list of 8.
In how many different ways can you choose the
books if order does not matter.
There are 56 combinations of 8 objects taking 5.
1
2
3
4
Combinations of 4 objects choosing 2
1
2
3
1
1
4
2
3
3
4
2
4
Each of the 6 groups represents a combination.
1
2
4
3
Permutations of 4 objects choosing 2
1
1
1
2
3
2
3
4
4
1
1
1
2
3
3
4
3
4
2
3
2
4
4
2
Each of the 12 groups represents a permutation.
The nCr and nPr key are the Combination
and Permutation keys on your
calculator.
To find them on the TI-83
First type in your n, then press
MATH across to PROB
Down to nCr or nPr
Example
In a race with eight horses, how many
ways can three of the horses finish in
first, second and third place? Assume
that there are no ties.
Example
The board of directors for a company has
twelve members. One member is the
president, another is the vice president,
another is the secretary and another is
the treasurer. How many ways can
these positions be assigned?
Distinguishable Permutations
The number of distinguishable
permutations of n objects where n1 are
one type and n2 are of another type and
so on, is:
n!
n1!•n2!•n3!•… nk!
Example
To find the number of permutations in the
word Mississippi
total letters
each set of repeats
=
11!
2!4!4!
Example
A contractor wants to plant six oak trees,
nine maple trees, and five poplar trees
along a subdivision street. If the trees
are spaced evenly apart, in how many
distinguishable ways can they be
planted?
Example
The manager of an accounting
department wants to form a threeperson advisory committee from the 16
employees in the department. In how
many ways can the manager do this?
Example
A word consists of one L, two E’s, two T’s
and one R. If the letters are randomly
arranged in order, what is the
probability the the arrangement spells
the word “letter”?
Example
A jury consists of five men and seven women.
Three are selected at random for an
interview. Find the probability that all
three are men.
Find the product of all the ways to choose
three men from five and 0 women from 7.
b. Find the number of ways to choose 3 jury
members from 12.
c. Divide a by b.
a.