Statistics 221

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Statistics 221

• Chapter 4 Part A
• Probability

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4.1 Experiments and Events

• In statistics, the word experiment means an
  event that generates an well-defined outcome.
• Examples of events flipping a coin (outcomes
  heads or tails) or drawing a card from a deck
  (outcomes face card or not), taking a test
  (outcomes pass or not), selecting one item from
  a batch of items (outcomes defective or not).
• Well use the word event rather than
  experiment when we refer to this phenomenon.

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Events

• Events can be single-outcome events or
  collective-outcome events.
• For example, if your event is flipping a coin,
  there is one single outcome that is heads or
  tails.
• But if your event is flipping 10 coins there
  are 10 outcomes which are collected into a
  single outcome such as the number of those 10
  flips that resulted in heads.
• Example this outcome HHHHTTHHTH would be
  considered an outcome of 7 for 7 heads.

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Sample space

• The sample space for an event is the set of all
  possible outcomes.
• A sample point is a member of the sample space,
  any one particular outcome.
• For example, if the event is flipping a coin,
  the sample space is heads, tails a sample
  point one particular outcome is heads.
• If your event is rolling two die, the sample
  space is 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-1, 2-2,
  etc.. To express the outcome as a single value,
  the sum of the face values on the two die could
  be used.

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Defining sample space

• When you want to define an events sample space,
  you determine how many possible outcomes there
  are and you (may be able to) list them all.
• For example, if your event involves tossing a
  coin one time, then the sample space is H, T -
  (heads or tails) and there are two possible
  outcomes.

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Defining sample space

• Sometimes, you can just list all the possible
  outcomes and count them. But some events have
  many, many possible outcomes too many to count
  one-by-one. If that is the case, there are
  short-cut formulas that can be used to calculate
  the number of possible outcomes.
• Combination events may have many possible
  outcomes and require the formula method to
  determine sample space.

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Counting the number of possible outcomes for
combination events

• If your event is a combination event such as
  flipping a coin two times, then the sample space
  is H, H, H, T, T, H, T, T (four possible
  outcomes).
• If your event is flipping a coin three times,
  then the sample space is H, H, H, H, H, T,
  H, T, H, H, T, T, T, H, H, T, H, T, T,
  T, H, T, T, T (eight possible outcomes).

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Counting possible outcomes for combination events

• So how many outcomes are possible if you toss a
  coin 4 times? 5?
• Recall that some events have many, many possible
  outcomes as with some combination events and
  there may be too many to list.
• Fortunately, there is a short-cut formula that
  can be used (if certain conditions are met) to
  calculate the number of outcomes in the sample
  space.

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The formula for counting the number of possible
outcomes

• If an experiment/event has (t) steps/phases,
  multiply the number of possible outcomes (n) at
  step 1 by the number of possible outcomes (n) at
  step 2 by the number of possible outcomes at
  step t.
• Example if the step is tossing a coin and
  there are two possible outcomes (n 2) and your
  event is tossing a coin 5 times (t5), then the
  number of possible outcomes is 2 2 2 2 2
  or 32.
• H, H, H, H, H would be one of those possible
  outcomes or in other words, one sample point.

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Applying the counting formula to define sample
space

• If you roll a die once (t1), the number of
  possible outcomes each time you roll is 6 (n6),
  so total number of possible outcomes is 6.
• The sample space is 1, 2, 3, 4, 5, 6.
• If you roll it twice (t2), the number of
  possible outcomes is 6 (n6) each time, then the
  total number of possible outcomes is 6 6 or
  36.
• The sample space is 1-1, 1-2, 1-3 .
• If you roll it three times (t3), the number of
  possible outcomes is 6 6 6 or 216.

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Example ofApplying the counting formula to
define sample space

• A telephone number has a three-digit area code.
• How many three digit area codes can there be?
• Each digit can be any digit from 0 to 9 (10
  possible outcomes).
• So if n10, t3, then there can be 10 10 10
  or 1,000 possible area codes.

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What if the number of possible outcomes are not
the same for each step?

• What if I said that the middle digit must be a 0
  or 1? Now, how many area codes can there be?
• Wed have to use this formula 10 2 10 200.
• (10 possibilities for the first digit, 2
  possibilities for the 2nd digit, 10 possibilities
  for the 3rd digit.)

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Another example

• The Kentucky Power and Light Company is about to
  start a large construction project. The project
  has two phases (1) design and (2) construction.
• The design phase could take 2, 3, or 4 months
  the construction phase could take 6, 7, or 8
  months.
• Define the sample space. In other words,
  determine how many possible outcomes there are
  there for completion times and list them.

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Example of defining sample space Kentucky Power
and Light Co.

• How many possible outcomes?
• 3 (for phase 1) 3 (for phase 2) 9

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What are the possible outcomes?
Possibilities for the number of months to complete

• Design Construction
• Phase 1 Phase 2 Total
• 2 6 8
• 2 7 9
• 2 8 10
• 3 6 9
• 3 7 10
• 3 8 11
• 4 6 10
• 4 7 11
• 4 8 12

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Example of defining sample space Bradley
Investments

• Bradley has invested in two stocks, Markley Oil
  and Collins Mining. Bradley has determined that
  the possible outcomes of these investments three
  months from now are as indicated in the table
  below. Define the sample space. In other words,
  how many possible outcomes are there and what are
  they?

Possible Profit in 3 Months (in 000)

• Markley Oil Collins Mining
• 10 8
• 5 -2
• 0
• -20

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How many possible outcomes are there?

• Markley n1
• Collins n2
• n1 n2 (4) (2) 8

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What are the outcomes?A tree diagram is
sometimes used to express the outcomes
Markley Oil Collins Mining
Gain 8
(10, 8) Gain 18,000 (10, -2) Gain
8,000 (5, 8) Gain 13,000 (5, -2)
Gain 3,000 (0, 8) Gain 8,000 (0,
-2) Lose 2,000 (-20, 8) Lose
12,000 (-20, -2) Lose 22,000
Lose 2
Gain 10
Gain 8
Lose 2
Gain 5
Gain 8
Even
Lose 2
Lose 20
Gain 8
Lose 2
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Defining sample space using the permutation
formula

• In the previous examples, when we defined sample
  space, we were actually asking the question How
  many permutations are possible?
• The permutations formula is written like this

N!
P
N

n
(N n)!

• where N is the number of outcomes for one step
  and n is the number of steps/times.

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Permutations

• The permutations formula can be simplified. For
  example, what is the number of possible 3-digit
  area codes that are possible?

10 10 10 1,000
10P3
N is 10 n is 3
(multiply 10 by itself 3 times.)
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Permutations with replacement

• The previous question was about defining the
  sample space when we were sampling with
  replacement.
• When you sample with replacement, each time you
  select a digit, there is still the same number of
  digits to select from the next time.
• For example, you have a choice of 10 possible
  values for the first digit, after you select the
  first digit, there are still 10 possible values
  for the 2nd and 3rd digits, etc.

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Permutations without replacement

• What if I said, the same digit cannot be used
  more than once? How many area codes are possible?
  Now the formula is

10P3
10 9 8 720

• This is called sampling without replacement.
  Each time you select a digit, there is one less
  digit to select from next time. The sample space
  will be reduced when you sample without
  replacement.

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Combinations

• Combination questions are similar to permutation
  questions but all different orders count as one
  possible outcome/sample point.
• For example, if we had to select three letters
  from the alphabet, the outcome of ABC is not a
  distinct outcome from CBA.
• The combinations formula is

N!
C
N

n
(N n)! n!

• where N is the number of outcomes for one step
  and n is the number of steps/times.

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A combinations example

• If you had to select three letters from the
  alphabet, how many different combinations of
  three letters are possible?
• The combinations formula can be simplified to

26 25 24 3 2 1
26C3
2600
Combination questions are usually considered as
without replacement situations.
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Example Combinations(order doesnt matter)

• There are nine candidates available for
  committees. How many different three-person
  committees are possible? (order doesnt matter,
  and selection is without replacement

NCn
9 8 7
84
3 2 1
N is 9 n is 3
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Example Permutations(order matters)

• There are nine people on the Board of Trustees.
• A president, vice president, and secretary must
  be chosen. How many different possible sets of
  president, VP, secretary are possible?
• Order matters, selecting without replacement.
• 9 8 7

NPn
504
N is 9 n is 3
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Practice question ( 2 on p. 149)

• How many ways can three items be selected from a
  group of six items? Use the letters A, B, C, D,
  E, and F to identify the items, and list each of
  the different combinations of the three items.
• Assume without replacement.

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Answer (2 on p.149)
NCn
6 5 4 3 2 1
120/6 20
N is 6 n is 3

• ABC ABD ABE ABF ACD ACE ACF
• ADE ADF AEF BCD BCE BCF BDE
• BDF BEF CDE CDF CEF DEF

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Practice question (3 on p. 149)

• 3. How many permutations of three items can be
  selected from a group of six items? Use the
  letters A, B, C, D, E, and F to identify the
  items, and list each permutation of items for
  B,D, and F.
• Assume without replacement.

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Answer (p. 149)
NPn
6 5 4
120
N is 6 n is 3

• ABC, ABD, ABE, ABF, ACB, ACD, ACE, ACF, ADB, ADC,
  ADE, ADF, AEB, AEC, AED, AEF, AFB, AFC, AFD, AFE,
  BAC BAD, BAE, BAF, BCA, BCD, BCE, BCF, BDE, BDF,
  BEF etc.
• In this case, ABC is a different sample point
  than ACB.

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How many combinations are there?

• To win the Pennsylvania lottery you have to guess
  six two-digit numbers and all six have to match
  the winning numbers. Each number can be from 01
  to 69 but no number will occur more than once in
  the 6-number winning set.
• As long as you select all 6 numbers correctly,
  order doesnt matter.

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How many combinations are there?

• You must select all six numbers correctly but
  they can be in any order and there are no
  duplicate numbers in the winning set.

NCn
69 68 67 66 65 64 6 5 4 3
2
N is 69 n is 6
119,877,472
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How many permutations are there?

• Singing legend Frank Sinatra recorded 381 songs.
  From a list of his top 50 songs, you must select
  three that will be sung in a medley as a tribute
  at the next MTV Music Awards Ceremony. The order
  of songs is important so that they fit together
  well. If you select three of Sinatras top 50
  songs, how many different sequences are possible?

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How many permutations are there?

• You have a choice of 50 songs, you must pick 3.
  You cant pick the same one twice (selecting
  without replacement), and order matters.

NPn
50 49 48 117,600
N is 50 n is 3
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Assigning Probabilities to Events
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Probability

• Probability is a numerical measure of the
  likelihood that an event will occur.
• Probabilities are expressed as percentages that
  range from 0 to 1.
• A probability near 0 indicates an event is very
  unlikely to occur.
• A probability near 1 indicates an event is almost
  certain to occur.
• A probability of 0.5 indicates the occurrence of
  the event is just as likely as it is unlikely.
• Probabilities can be used to express the degree
  of uncertainty of an event.

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Probability as a Numerical Measureof the
Likelihood of Occurrence
Increasing Likelihood of Occurrence
0
1
.5
Probability
Definitely WILL happen
Definitely WILL NOT happen
The occurrence of the event is just as likely
as it is unlikely.
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Basic requirements for assigning probabilities to
events

• A probability assigned to a particular outcome
  must be between 0 and 1 inclusively
• 0
• The probabilities assigned to each possible
  outcome of an event must sum to 1.0
• P(E1) P(E2) P(E3).. P(En) 1

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Three methods for assigning probabilities to
events

• Classical Method
• Assigning probabilities based on the assumption
  of equally likely outcomes.
• Relative Frequency Method
• Assigning probabilities based on experimentation
  or historical data.
• Subjective Method
• Assigning probabilities based on the assignors
  judgment.

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Classical method

• If an experiment has n possible outcomes, this
  method would assign a probability of 1/n to each
  outcome.
• Example
• Experiment Rolling a die
• Sample Space S 1, 2, 3, 4, 5, 6
• Probabilities Each sample point has a 1/6
  chance of occurring.

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What is the probability?

• If you flip a coin, what is the probability of
  getting 5 heads?
• Since there are 25 possible outcomes (in the
  sample space) and HHHHH is one sample point,
  there is a 1/32 probability of getting that
  sample point is
• 1/32 .03125 or 3.125

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What is the probability?

• If you draw a card from a deck, what is the
  probability that it will be an ace?
• Since there are 52 cards and 4 aces, the
  probability is 4/52 or 1/13 or .0769 or 7.69.

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What is the probability?

• If there are 1,000 possible area codes, and one
  area code is selected randomly, what is the
  probability of getting 901?
• 1 / 1000 or .001 or .1

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What is the probability?

• If you had to select any three letters from the
  alphabet, what is the probability of getting ABC?
• 1 / 2600 or .000385 or .0385

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What is the probability of winning the
Pennsylvania lottery?

• If you buy one ticket, it is
• 1 / 119,877,472 or .00000000832

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Relative Frequency Method

• Probabilities are based on past observance of how
  often an event occurs. We assume that past
  frequency will continue into the future.
• Example
• P(rolling a 6) of times a 6 occurred
• of times the dice
  was rolled
• Example 115 / 760 ? 16.34

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Calculating probabilities using the relative
frequency method

• Last year, American Airlines booked 897,342
  seats. 26,920 of those passengers (3) didnt
  show up for their flight. Therefore, the
  probability that a passenger wont show up for
  his flight is 3.
• During trial tests, 100,000 patients tried a new
  cholesterol-reducing drug and 7,000 of them had
  an adverse reaction. Therefore, the probability
  that a person will have an adverse reaction to
  the drug is 7.

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Another Example of the Relative Frequency Method
Lucas Tool Rental

• Lucas would like to assign probabilities to the
• number of floor polishers it rents per day.
  Office
• records show the following frequencies of daily
  rentals
• for the last 40 days.
• Number of Number
• Polishers Rented of Days
• 0 4
• 1 6
• 2 18
• 3 10
• 4 2

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Calculating probabilities using the relative
frequency method

• The probability that exactly one polisher will be
  needed is 15.
• Number of Number
• Polishers Rented of Days Probability
• 0 4 .10 4/40
• 1 6 .15 6/40
• 2 18 .45 etc.
• 3 10 .25
• 4 2 .05
• 40 1.00

The probability assignments are given by dividing
the number-of-days frequencies by the total
frequency (total number of days).
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Subjective probability method

• When economic conditions and a companys
  circumstances change rapidly it might be
  inappropriate to assign probabilities based
  solely on historical data.
• We can use any data available as well as our
  experience and intuition, but ultimately a
  probability value should express our degree of
  belief that the experimental outcome will occur.
• The best probability estimates often are obtained
  by combining the estimates from the classical or
  relative frequency approach with the subjective
  estimates.

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Example where the subjective probability method
was used

• The weatherman says there is a 98 chance of
  rain.
• Scientists predict that there is a .0005 chance
  that a large meteorite will hit the earth.
• President Bush says there is a 65 chance that
  the social security administration fund will be
  bankrupt by the year 2020.

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Homework 6

• 9 on page 150
• Define sample space in terms of the of possible
  outcomes
• 10 on page 150
• What is the probability?
• 11 on page 150-1
• What is the probability?
• 12 on page 151
• Define sample space in terms of the of outcomes
  and assign probabilities to outcomes