Basic Probability

1
Basic Probability

• Introduction
• Sample Spaces and Events
• Probability Models
• Basic Theorems of Probability

2
Materials for Review and Practice

• Student Notebook
• Slides 8 thru 27 (pps. 21-31)
• Supplemental Texts
• Anton, H. Kolman, B and Averbach, B (1992)
  Applied Finite Mathematics, 5th Ed, Orlando, FL
  Saunders College Publishing Sections 7.1, 7.2,
  7.3
• Student Manual (pps. 33-50)

3
Laws of Chance

• In 1952, a New York gambler known as Fat the
  Butch gave even-money odds that in 21 rolls of a
  pair of dice he would get at least one double-6.
  In a series of bets with a gambler known as The
  Brain, Fat the Butch lost 50,000.
• Did The Brain know something that Fat the Butch
  should have known? Over the next few weeks you
  will be able to answer that question.

Source Orkin, M. (2000). What Are The Odds?
W.H. Freeman and Company, N.Y.
4
Theoretical versus Empirical

• Theoretical probabilities are those that can be
  determined purely on formal or logical grounds,
  independent of prior experience.
• Empirical probabilities are estimates of the
  relative frequency of an event based by our past
  observational experience.

5
Theoretical Probability

• Toss a coin and determine the probability of
  observing a heads
• Assume that the coin is balanced.
• What are the possible outcomes?
• What is the probability of heads?
• Draw a card from a standard 52 card deck. What is
  the probability of drawing a spade?
• Assume that the cards are sufficiently
  randomized.
• What are the possible outcomes?
• What is the probability of spades?

6
Empirical Probability

• Consider the previous example of tossing a
  balanced coin. Suppose instead we had reason to
  believe that the coin was not balanced (e.g., we
  notice that the coin is slightly bent). How
  might we determine the probability of this
  biased coin?
• Solution Toss the coin many, many times and
  record the frequency of heads vs. tails.

7
Empirical Probability

• We can subdivide empirical probabilities into
  two categories
• Objective versus Subjective
• Objective probabilities are those that are based
  on observations of past occurrences of events,
  under what are hopefully the same conditions that
  currently prevail (as in our example of the
  biased coin).

8
Subjective Probabilities

• Many processes are not repeatable independently
  under identical conditions. They are
• Empirical in the sense that they are ultimately
  based on past observation
• Subjective in the sense that the particular
  observation(s) upon which the particular
  probability estimate(s) are based, is not well
  defined, that is, a independent observer could
  not be instructed on how to arrive at the same
  probability

9
Example (Subjective Probability)

• A stockbroker says there is a 70 chance that IBM
  will go up at least 10 points in the next month.
  The brokers probability is based on a careful
  study of market data. Although current market
  trends may be similar to past situations, the ups
  and downs of IBM are not repeatable under
  identical conditions like the heads and tails of
  coin tossing.

10
More Examples of Empirical Probability

• Subjective Probabilities
• What is the probability that upon graduation, you
  will be offered a position on your first job
  interview?
• What is the probability that you will be earn an
  A on your first test this semester?

• Objective Probabilities
• What is the probability of a certain automobile
  insurance applicant filing a claim?
• What is the probability that a certain production
  process will produce a defective flashbulb?

11
Basic Probability

• Sample Spaces and Events
• I hope I break even this week. I need the
  money.
• - Veteran Las Vegas gambler

12
Relative Frequency

• Toss a coin and note which side lands up it is
  impossible to predict the outcome (heads or
  tails) in advance with certainty
• Toss coin again and again, the proportions of
  heads and tails will tend to a fixed value (we
  expect 0.5 if the coin is perfectly balanced).
• To generalize, suppose an experiment can be
  repeated indefinitely under fixed conditions and
  suppose that during n repetitions a certain event
  occurs with frequency f
• We call the ratio f / n the relative frequency
• If f / n approaches a fixed value, we call that
  value the probability of the event.

13
Summary of 20,000 Coin Tosses
14
Law of Large Numbers

• The more repetitions, the better the
  approximation p ? f / n.
• This is sometimes referred to as the Law of Large
  Numbers, which states that if an experiment is
  repeated a large number of times, the relative
  frequency of the outcome will tend to be close to
  the probability of the outcome.
• Shortly, we will consider another approach to
  determining the probability of an event based on
  logical reasoning.

15
Probability Experiment

• Throughout this course we will be concerned with
• Experiments whose outcomes cannot be predicted in
  advance with certainty
• The outcomes themselves
• The term experiment is used in a broad sense to
  mean an observation of any physical occurrence.

16
Probability Experiments

• Whenever we manipulate or make an observation of
  our environment with an uncertain outcome, we
  have conducted a probability experiment.
• Examples
• Taking an exam
• Playing poker
• Delivering a sales pitch
• Testing automobile shock-absorbers

17
Sample Space

• The set of all possible outcomes of an experiment
  is called the sample space for the experiment.
• The outcomes in the sample space are called
  sample points
• The sample points and the sample space depend on
  what the experimenter chooses to observe.

18
Example Toss a Coin Twice

• We could choose to record the sequence of heads
  (H) and tails (T), then
• S HH, HT, TH, TT
• We could choose to record the total number of
  tails observed, then
• S 0, 1, 2
• We could choose to record whether the two tosses
  match (M) or do not match (D), then
• S M, D

19
Exercise (Sample Spaces)

• Determine the sample space of the following
  experiments
• Toss a die and record the number on the top face
• Turn on a light and record if bulb is burned out
• Observe General Electric common stock and record
  whether it increased (i), decreased (d) or
  remained unchanged (u) during one market day
• Record the sex of successive children in a
  three-child family

20
Events

• Events are sets
• An event, E, is a subset of the sample space and
  it denoted by
• An event E is said to occur if the outcome of an
  experiment is an element of E
• Events are classified as either simple or
  compound.

21
Experiment Toss a die once and record the number
on the top face.
The sample space, S 1, 2, 3, 4, 5, 6 Some
events associated with this experiment
22
Simple Events vs. Compound Events

• A compound event is any event that can be
  decomposed into other events.
• E3 , E4 and E5 are compound events
• A simple event cannot be decomposed.
• E1 and E2 are simple events

23
Exercise

• Consider the experiment of flipping a balanced
  coin three times and recording the sequence of
  heads (H) and tails (T).
• Using a tree diagram determine the sample space
  for the experiment
• List two events that correspond to this experiment

24
Some Special Events

• Note that if S is a sample space of some
  experiment, then both S and the empty set are
  subsets of S and are therefore events defined on
  S.
• In any experiment, the event S must occur
  therefore it is called a certain event
• The empty set contains no sample points therefore
  it cannot occur. Such an event is called an
  impossible event
• Mutually exclusive events are events that cannot
  both occur at the same time. Symbolically, E and
  F are mutually exclusive if

25
Mutually Exclusive Events

• An experimenter tosses a die and records the
  number on the top face.
• Let E be the event that the number is even, then
• E 2, 4, 6.
• Let O be the event that the number is odd, then
• O 1, 3, 5.
• Since , the events are mutually
  exclusive (they cannot both happen at the same
  time).

26
Important Terminology

• Experiment
• Sample space
• Outcomes
• Sample points (Simple or Elementary Events)
• Probability model
• Events
• Certain event
• Impossible event
• Mutually exclusive events

27
Comparing The Language of Set Theory with
Probability Theory

• Events are sets whose elements are sample points.
• Mutually exclusive events are disjoint sets
• The sample space of an experiment is the
  universal set
• All set operations apply, that is, set unions,
  intersections, complements, DeMorgans laws etc.

28
Basic Probability

• Probability Models for Finite Discrete Sample
  Spaces
• Basic Theorems of Probability

29
Finite Sample Spaces

• Experiments which have finitely many outcomes are
  said to have finite sample spaces.
• In Project 1, we will be concerned with finite
  sample spaces.

30
Fundamental Properties

• The relative frequency of the sample space must
  be 1
• Negative relative frequencies do not make sense
• If two events are MUTUALLY EXCLUSIVE, the
  relative frequency of their union must be the sum
  of their relative frequencies.
• If the events E1, E2, , En are pair-wise
  mutually exclusive then

31
Probability Models

• Consider an experiment with
• Ss1, s2, , sn
• When probabilities are assigned to the elementary
  events of the experiment so that Property 1 and 2
  hold, we call that assignment a probability model
  for the experiment.

32
Example of a Probability Model

• A six-sided die is tossed and the number on the
  top face is recorded. Then
• S1, 2, 3, 4, 5, 6
• Assume that the die is symmetric and perfectly
  balanced then
• Since the probabilities of the elementary events
  must add up to 1,
• and

33
Exercise

• Consider an experiment of tossing a loaded die
  where it is known that 1, 3, and 5, have the same
  chance of occurring, whereas each of 2, 4, and 6
  is twice as likely to occur as 1.
• Construct a probability model for the experiment
  and use your model to determine the probability
  of the event E, you observe a number less than 4.

34
Experiments with Equally Likely Outcomes

• If an experiment can result in any one of k
  equally likely outcomes and if an event E
  contains m sample points, then the probability of
  the event E is

• Example 1. Consider the experiment of tossing a
  fair die and observing the number on the top
  face. Find the probability of
• The event E a even number of tossed
• The event G a number divisible by 3 is tossed
• Example 2. A batch of 7 resistors contains 2
  defectives. If a resistor is selected at random,
  what is the probability that it is defective?

35
Five Steps in Calculating P(E)

• Define the experiment and clearly determine how
  to describe one simple event.
• List the simple events associated with the
  experiment and test each to be certain that they
  cannot be decomposed. This defines the sample
  space S.
• Assign probabilities to the sample points in S
  making certain that the Fundamental Properties
  for a discrete sample space are preserved.
• Define the event, E, as a specific collection of
  sample points.
• Find P(E) by summing the probabilities of the
  sample points in E.

36
Example Three

• A balanced coin is tossed three times.
• Let E1 be the event that you observe at least two
  heads. What is P(E1)?
• Let E2 be the event that you observe at exactly
  two heads. What is P(E2)?
• Let E3 be the event that you observe at most two
  heads. What is P(E3)?
• Are E1 and E3 mutually exclusive?

37
Basic Theorems of Probability

• Let S be a finite discrete sample space and let E
  and F be events defined on S
• Theorem 1 P(?)0, where ? is the empty set.
• Theorem 2 For any two events E and F in S,
• P(E ? F) P(E) P(F) - P(E ? F)
• Theorem 3 If E is an event in S, then
• P(EC) 1 - P(E)