Questions

1
Questions

• The syllabus.
• The homework assignment.
• Yesterdays lab.
• Anything else you want me to answer.

2
Sections 2.1 thru 2.3

• Basic Properties of Probability
• Counting Techniques

3
Review Vocabulary

• Experiment
• Deal 12 cards to group A from deck R1-R12 and
  B1-B11.
• Sample Space (S) The collection of all possible
  outcomes
• The list of all hands group A could get.
• Event a subset of the sample space S.
• An event is simple if it contains one outcome.
• An event is compound if it contains more than one
  outcome.
• Group A gets dealt R1- R3, and B1-B9 is a simple
  event (assuming that we are disregarding order).
  While Group A gets 3 reds and 9 blacks is a
  compound event.
• Today probability of events.

4
Union, Intersection, and Complements

• Union. A ? B contains those
  outcomes in A, B, or both.
• Intersection. A ? B contains those
  outcomes that are in both A and B.
• Complement. A? contains all elements
  not in A.

5
Definition of Probability

• Given an experiment with outcome space S, a
  probability is a function P which assigns a real
  number P(A) to each event A ? S, subject to the
  following rules
• for all A ? S, P(A) ? 0,
• P(S) 1,
• given mutually exclusive events A1, A2, A3, ... ,
  P(A1 ? A2 ? A3 ? ... ) P(A1)
  P(A2)
• These properties guarantee that all probabilities
  are between 0 and 1, as well as some other useful
  properties that we will look at in a few slides
  slides.

6
Interpretations of probability.

• Objective probability.
• The objective interpretation of probability is
  based on the notion of limiting frequency and is
  thus limited to experimental situations that are
  repeatable.
• Subjective probability.
• The language of probability is often used in
  situations that are unrepeatable. In these cases
  probability assignment is subjective.
• For example, there is a _______ chance Hope will
  beat _____ Saturday.
• The mathematical development is limited to
  objective probability and caution should be taken
  when applying results to subjective
  probabilities.

7
Interpreting Objective Probabilities

• The probability function models the actual
  experiment well when the probability gives the
  proportion of times the event would occur in a
  very long series of repetitions.
• Empirical probability refers to the number
  obtained from simulation while theoretical
  probability refers to the number obtained from a
  model.

8
Properties of Probability

• For each A ? S, P(A) P(A?) 1.
• When events A and B have no outcomes in common,
  they are said to be mutually exclusive or
  disjoint. For disjoint events, P(A ? B)0.
• Given A, B ? S, P(A ? B) P(A) P(B)
  ? P(A ? B).
• Use Venn Diagrams Frequently

9
Example 1.

• A person is applying for two jobs. They feel
  that the probability they will get an offer for
  the first job is 0.6, the probability they will
  get an offer for the second job is 0.8, and the
  probability they will get both offers is 0.5.
• Is this subjective or objective?
• Calculate the probability they will get at least
  one job offer.
• Calculate the probability they will get no job
  offers.

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Example 2

• A certain system can have three different types
  of defects. Let Ai denote the event that the
  system has defect i. Suppose that P(A1)0.12,
  P(A2)0.07, P(A3)0.05, P(A1?A2)0.06,
  P(A1?A3)0.03, P(A2?A3)0.02, and P(A1?A2
  ?A3)0.01.
• Is this subjective or objective?
• What is the probability that a system does not
  have a defect?
• What is the probability that a system has at
  least one defect?
• What is the probability that a system has at
  least two defects?
• What is the probability that a system has all
  three defects?

11
Counting and Probability

• Suppose a sample space S contains finitely many
  elements, and assume that each of these outcomes
  is equally likely.
• In this case, given any event A ? S, we can
  calculate the probability that A occurs in this
  way number of elements in
  A P(A) ???????????
  number of elements in S

12
Multiplication Principle and Tree Diagrams
If there are n1 ways to perform Task 1, n2
ways to perform Task 2, . . . nm ways to
perform Task mthen the total number of ways of
performing the sequence of tasks from 1 to m is
the product n1n2...nm. Example 3 How many 1
topping pizzas can be made if you have 3 choices
for size, 2 choices for crust type, and 12 pizza
toppings to choose from? How many pizzas can be
made if you can include any number of toppings?
13
Permutations.
Definition. Any ordered sequence of k objects
taken from a set of n distinct objects is called
a permutation of size k of the objects. The
number of permutations of size k that can be
constructed from the objects is denoted by Pn,k.
In lab yesterday, we showed Pn,k n (
n ? 1 ) ( n ? 2 ) ... ( n ? k 1) Example 4. If
there are 12 players on a little league baseball
team, how many ways can the 9 field positions be
assigned?



14
Combinations
Definition. Any un-ordered subset of k objects
taken from a set of n distinct objects is called
a combination of size k taken from the objects.
The number of combinations of size k that can be
constructed from the n objects is denoted
by Example 5. A committee of 10 would like to
create a subcommittee of 3. How many different
ways can this be done?
15
Hypergeometric Probabilities

• Population of size N
• with M successes and N-M failures.
• You take a sample of size n.
• The probability that you will have exactly k
  successes in your sample of size n is
• (Logic)
• (Formula)

16
Example 6.

• Fifteen telephones have just been received at an
  authorized service center (5 cellular, 5
  cordless, and 5 corded). Suppose that the phones
  are serviced in random order.
• What is the probability that all of the cellular
  phones are among the first 6 to be serviced?
• What is the probability that after servicing 6
  phones, phones of only two of the three types
  remain to be serviced?
• What is the probability that after servicing 10
  phones, phones of only two of the three types
  remain to be serviced?
• What is the probability that two phones of each
  type are among the first 6 to be serviced?

17
Summary.

• Section 2.2 From the probabilities of some
  events, find probabilities of other events using
  properties and/or Venn diagrams.
• Section 2.3 Break the experiment down into
  equally likely outcomes. Probability of an event
  is the number of outcomes in the event divided by
  the number of possible outcomes.
• Do homework problems 12, 13(not f), 16, 18, 22,
  23, 30, 32, 33, 34, 38 by Tuesday.