Chapter 3: Probability

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Chapter 3 Probability
2
Section 3-1 Introduction

• The decision to work out or foreclose a loan
  depends on the probability of success or failure
  of the workout. This will determine how much
  money we will recover from the loan.

3
Definitions

• A Trial is an activity where the result is
  unknown. (this is sometimes called an experiment,
  or a random experiment)
• An Outcome is one specific result of a trial.
• A Sample Space is the set of all possible
  outcomes. It is usually represented by a capital
  S, but we will use the symbol to represent
  sample space. (S is needed for something else)
• A Probability is the proportion of times a
  particular event will occur. It is usually
  represented by a capital P.

4

• Flip a coin 10 times.
• Record the number of times a head comes up.
• Add up the total number for the whole class
• What do you observe?
• How many heads out of number of flips?

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• Flipping a coin is a trial
• A H or T are outcomes of the trial.
• Sample Space H, T
• P(H) ½
• P(T) ½

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Example

• You have three nickels (coins) that you flip into
  the air and onto the table.
• A trial would be flipping the three coins.
• One outcome would be heads-tails-heads (HTH for
  short). What are the other possible outcomes?
  (see next slide)
• An event is a subset of outcomes from a sample
  space

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Example

• The sample space for this trial of flipping 3
  coins (8 in this case.)
• HHH HHT
• HTH HTT
• TTT TTH
• THT THH

8
Some Notation

• Suppose we are interested in knowing all the
  possible outcomes where we get two heads and one
  tail. We will call this event A
• We can write A in set notation
• A HHT, HTH, THH
• There are therefore 3 ways this can happen out of
  a total of 8 possible outcomes. We say the
  probability of the event A happening is 3/8.

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• For example, when rolling one die,
  1,2,3,4,5,6.
• Let the event E rolling an even number. Then
  the set E is all the ways to roll an even number
  E 2,4,6. This is a subset of .

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Theoretical Probability

• Let E be an event. Then
• P(E) k/n
• k the number of ways event E can occur
• n the total number of possible outcomes
• CAUTION This formula is only valid if each
  outcome is equally likely.

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Example

• A marble is drawn from a bag. There are 15 red,
  12 yellow, and 18 blue marbles in the bag.
• What is the probability of randomly drawing a
  single red marble from the bag?
• What is the probability of randomly drawing a
  single blue marble from the bag?

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Question for Discussion (something to think
about. But we will discuss later)

• A marble is drawn from a bag. There are 15 red,
  12 yellow, and 18 blue marbles in the bag.
• What is the probability of drawing a red marble
  from the bag, setting it aside, and then
  immediately drawing a second red marble?
• Does the result change if you replace the first
  red marble before drawing the second time? If so,
  why?

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Empirical Probability

• Empirical data is that which you observe. For
  example, you have collected data that indicates
  that of the last 550 loans a bank granted, 42 of
  them were foreclosed upon.
• Then based on the empirical data, you might say
  that the probability of a loan going into
  foreclosure is 42/550.

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Empirical Probability

• Let E be an event. Then
• P(E) k/n
• k the number of times event E has occurred in
  the past (under similar circumstances)
• n the number of trials in the past

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Law of Large Numbers

• Empirical probabilities are basically just
  estimates. They do not necessarily predict the
  outcome of a particular trial. (The next loan
  will go to foreclosure!) We do know this,
  however The outcome of one trial cannot be
  predicted, but, one can predict what will happen
  over a series of many trials.

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3-2 Combining Events

• A marble is drawn from a bag. There are 15 red,
  12 yellow, and 18 blue marbles in the bag.
• What is the probability of randomly drawing
  either a yellow or blue marble from the bag?
• What about What is the probability of not
  picking a red marble?
• How are these related to the topics we discussed
  in Chapter 2?

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3-3 Dice Problem

• Suppose you have two dice that you roll onto a
  table. Here is the sample space.

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Questions

• What is the probability of getting a total of 10?
• What is the probability of getting the same
  number on each die?
• What is the probability of getting a prime number
  total?
• What is the probability of NOT getting a total of
  7?
• What is the probability of getting either a total
  of 8 or a total of 11?
• What is the probability of getting either a 4 or
  an odd number on one of the dies?
• What is the probability of getting a 4 and an odd
  number on the other die?

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• Rolling a die
• Let event A an even number
• Event B an odd number
• Note that A and B are disjoint
• We call these events mutually exclusive events
  when they dont have any outcomes in common.

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3-5 Rules of Probability

• Let A and B be events, and let be the sample
  space.
• Rule 1 0 ? P(A) ? 1
• Rule 2 P() 1
• Rule 3 P(AC) 1 P(A)
• Rule 4 If A and B are mutually exclusive events,
  then
• P(A ? B) P(A) P(B)

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Mutually exclusive versus Independent events

• When you roll a die, the events odd and even
  are mutually exclusive because they cannot happen
  at the same time.
• Hence, P(odd or even) P(odd) P(even)
• When you roll a die two times, the events that
  the 1st die is odd and the 2nd die is odd are
  independent events. The events does not affect or
  influence each other.
• Hence, Probability P(odd) P(odd) 3/63/6

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Not mutually exclusive events

• Roll a die.let us define
• C number that comes up is 1 or 2 or 3
• D number that comes up is 3 or 4 or 5
• Hence, C and D are not mutually exclusive
• P(C or D) P(C) P(D) - P(C and D)
• 3/63/6 - 1/6 5/6

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Rolling two dice

• What is the probability of getting a 4 and an
  odd number on the other die?
• Independent events
• hence, 1/6 3/6 3/36
• (4,1) (4,3), (4,5)

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• A marble is drawn from a bag. There are 15 red,
  12 yellow, and 18 blue marbles in the bag.
• What is the probability of drawing a red marble
  from the bag, setting it aside, and then
  immediately drawing a second red marble? (not
  independent events)
• Does the result change if you replace the first
  red marble before drawing the second time? If so,
  why? (independent events)

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• (a) 15/45 14/44
• (b) 15/45 15/45
• NOTE Probability that the 2nd marble is red
  knowing that the 1st marble is red and it is not
  replaced is just 14/44

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3-6 Venn Diagrams

• A total of 70 students are randomly interviewed.
  23 own a car. 45 own a bike. 18 own both a car
  and a bike. Draw a Venn diagram that displays all
  of the probabilities related to this survey.

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• Find the probabilities
• A student randomly chosen owns a car but not a
  bike.
• A student randomly chosen does not own either a
  bike or a car.
• A student randomly chosen owns a car or a bike.
• A student randomly chosen owns a car and a bike.

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3-6B General Probability Formula

• If A and B are events, then
• P(A ? B) P (A) P (B) P (A ? B)
• Subtracting compensates for the double-counting
  error.

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• Suppose 8 of a certain batch of calculators have
  a defective case, and that 11 have defective
  batteries. Also, 3 have both a defective case
  and defective batteries. A calculator is selected
  from the batch at random. Find the probability
  that the calculator has a good case and good
  batteries.

30

• Ms Bezzone invites 10 relatives to a party her
  mother, 2 uncles, 3 brothers, and 4 cousins. If
  the chances of any one guest arriving first are
  equally likely, find the probabilities
• The first guest is an uncle or a cousin.
• The first guest is a brother or a cousin.
• The first guest is an uncle or her mother.

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The table shows the probability of a person
accumulating credit card charges over a 12-month
period
Charges Probability
Under 100 0.31
100-499 0.18
500-999 0.18
1000-1999 0.13
2000-2999 0.08
3000-4999 0.05
5000-9999 0.06
10000 or more 0.01
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Find the probability that a persons total
charges during the period are

1. 500 or more
2. Less than 1000
3. 500 to 2999
4. 3000 or more

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Questions

• For the following problems, the trial is rolling
  two dice ( a red and a green die) (be sure to
  avoid double-counting)
• What is the probability of sum of both dice being
  7?
• What is the probability that the red die will
  show an odd number or the sum of the two dice
  will be 8?
• What is the probability that the green die is 6
  or the sum of the two dice is 10?
• What is the probability that the green die shows
  an even number and the sum of the two dice is 10?

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Group Exercise

• See 36 from Chapter 3
• From a survey involving 1,000 people in a certain
  city, it was found that 500 people had tried a
  certain brand of diet cola, 600 had tried a
  certain brand of regular cola, and 200 had tried
  both types of cola. (Barnett p. 414)

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Group Exercise

• Draw and label a Venn Diagram that demonstrates
  this information in sets. Then try another Venn
  diagram using the probabilities.
• What is your sample space?
• Find the probability that a randomly selected
  person from the city has tried both of the colas.
  Find the probability that a randomly selected
  person from the city has tried the diet cola but
  not the regular cola.
• Find the probability that a randomly selected
  person from the city has tried the regular cola
  but not the diet cola.

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Group Exercise

• Find the probability that a randomly selected
  person from the city has tried neither of the
  colas.
• Find the probability that a randomly selected
  person from the city has tried either the diet or
  the regular cola. Try to see if you can compute
  this in TWO different ways.
• Find the probability that a randomly selected
  person from the city has tried one of the colas
  but not both. Write your answer in probability
  notation. Be careful on this onea picture should
  help.

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Focus on the Project

• Let S be the event that an attempted work out is
  successful and let F be the event that it fails.
  Use the COUNTIF function to find the fraction of
  past work outs which were successful. This
  fraction is our estimate for P(S). Likewise, we
  find the fraction of attempts that failed and use
  this as our estimate for P(F).

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Focus on the Project

• Example 1
• Use the Loan Records.xls to estimate P(S).
• DCOUNT will work
• COUNTIF will also do the job

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Focus on the Project

• Go to the section titled Project 1 Specifics
  (Chapter 7) and do Part 2a only.
• Also do Chapter 3 Focus On Project Memo.
• Edit your Written Report to reflect your newest
  information. Be sure to use proper probability
  and mathematics notation in your writing (use the
  Equation Editor to format all mathematical text).