Probability

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Probability
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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
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Example
Probability experiment
Roll a die
Sample space
1 2 3 4 5 6
Event
Die is even 2 4 6
Outcome
4
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Practice

• Use a tree diagram to develop the sample space
  that results from rolling two six-sided dice.

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Tree Diagrams
Two dice are rolled. Describe the sample space.
Start
1st roll
1
2
3
4
5
6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
1 2 3 4 5 6
2nd roll
36 outcomes
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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
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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?

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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?

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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 / (327) 3/12 1/4

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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

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Empirical
Probability is based on observations or
experiments.
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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?

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Empirical examples

• What is the probability of getting a bluegill?
• What is the probability of not getting a crappy?

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Subjective

• Subjective probability results from educated
  guesses, intuition and estimates.
• A doctors prediction that a patient has a 90
  chance of full recovery
• A business analyst predicting an employee strike
  being 0.25

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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.

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Probability

• If P(E) 0, then event E is impossible.
• If P(E) 1, then event E is certain.
• 0 ? P(E) ? 1
• Impossible Even Certain
• 0 .5 1

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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.
P(E) 1 - P(E)
E
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Example
The days 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
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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?

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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

• Properties of a probability distribution
• Each probability must be between 0 and 1,
  inclusive.
• The sum of all probabilities is 1.

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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
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Example
Decide whether each distribution is a probability
distribution. Explain your reasoning. x 5 6 7 8
P(x) 1/16 5/8 ¼ 3/16
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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
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Odds

• Odds for an event A are P(A)
• P(not A)
• In sports we often look at wins over losses.

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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.

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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
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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.
• Two dice are rolled.
• A first is a 4 and B second is a 4

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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
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Multiplication Rule
Two dice are rolled. Find the probability both
are 4s.
A first die is a 4 and B second die is a 4.
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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.

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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).
A or B
A and B
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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
Mutually exclusive
B
P(A and B) 0
When event A occurs it excludes event B in the
same trial.
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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
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Examples

• Determine whether the events are mutually
  exclusive or not.
• Roll a die
• A Roll a 3 B Roll a 4
• Select a student
• A select a male student B select a nursing
  major
• Select a blood donor
• A donor is type O B donor is female

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Examples

• Select a card from a standard deck
• A the card is a jack
• B the card is a face card
• Select a student
• A the student is 16 years old
• B the student has blue eyes
• Select a registered vehicle
• A the vehicle is a Ford
• B the vehicle is a Toyota

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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
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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)
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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
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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
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Contingency Table
The results of responses when a sample of adults
in 3 cities was asked if they liked a new juice
is
Omaha
Seattle
Miami
Total
Yes
100
150
150
400
No
125
130
95
350
Undecided
75
170
5
250
Total
300
450
250
1000
One of the responses is selected at random. Find
3. P(Miami or Yes) 4. P(Miami or Seattle)
1. P(Miami and Yes) 2. P(Miami and Seattle)
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Contingency Table
Omaha
Seattle
Miami
Total
Yes
100
150
150
400
No
125
130
95
350
Undecided
75
170
5
250
Total
300
450
250
1000
One of the responses is selected at random. Find
1. P(Miami and Yes) 2. P(Miami and Seattle)
250/1000 150/250 150/1000 0.15
0
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Contingency Table
Omaha
Seattle
Miami
Total
Yes
100
150
150
400
No
125
130
95
350
Undecided
75
170
5
250
Total
300
450
250
1000
3 P(Miami or Yes) 4. P(Miami or Seattle)
250/1000 400/1000 150/1000 500/1000 0.5
250/1000 450/1000 0/1000 700/1000 0.7
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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.
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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,
dont forget to subtract the probability of both
occurring.
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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.

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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

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Example

• X P(X)
• 0 .25
• 1 .50
• 2 .15
• 3 .10
• 0(.25) 1(.5) 2(.15) 3(.10) 1.1

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Example

• A bus arrives at a bus stop at noon, 1220 and
  100. You arrive at the bus stop at random times
  between noon and 100 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 1220? What is
  the mean wait time in that case?

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• What is the probability that you will arrive at
  the bus stop between 1220 and 100?
• What is your mean wait time in that case?

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1. Overall, what is your expected waiting time for
   the bus?
2. Would you expect your waiting time to be longer
   or shorter if the bus arrived at equally spaced
   intervals (say noon, 1230 and 100)?

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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?

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• 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.

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Example

• You are given 10 to 1 odds against rolling a
  double number (for example two 1s or two 2s)
  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?

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Terms

• The gamblers 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.

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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.

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• 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 gamblers
  fallacy?

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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.
• What is the expected value to you of a single
  game?
• What is the house edge?

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• If you played 100 games of blackjack in an
  evening, betting 1 on each hand, how much would
  you expect to win or lose?
• If you played 100 games of blackjack in an
  evening, betting 5 on each hand, how much would
  you expect to win or lose?

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• d. If patrons bet 1,000,000 on blackjack in one
  evening, how much should the casino expect to
  earn?

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7E

• Counting and Probability

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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!

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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?

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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?

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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
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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.
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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
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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.
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2
4
3
1
Combinations of 4 objects choosing 2
4
1
2
3
4
3
2
4
Each of the 6 groups represents a combination.
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2
4
3
1
Permutations of 4 objects choosing 2
4
4
1
1
2
2
3
3
4
4
3
3
2
2
4
4
Each of the 12 groups represents a permutation.
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• 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

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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.

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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?

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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!

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Example

• To find the number of permutations in the word
  Mississippi
• total letters 11!
• each set of repeats 2!4!4!

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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?

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Example

• The manager of an accounting department wants to
  form a three-person advisory committee from the
  16 employees in the department. In how many ways
  can the manager do this?

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Example

• A word consists of one L, two Es, two Ts and
  one R. If the letters are randomly arranged in
  order, what is the probability the the
  arrangement spells the word letter?

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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.
• Find the number of ways to choose 3 jury members
  from 12.
• Divide a by b.