CAS MA 115 Lecture Notes - Lecture 5: Sample Space, Regional Policy Of The European Union, Conditional Probability
1 May 2018
School
Department
Course
Professor
CHAPTER 5 – PROBABILITY
Section 5.1 – Probability Rules
Objective 1 – Introduction to Probability
• Probability – a measure of the likelihood of a random phenomenon/chance/behavior
occurring
o Meant to describe the long-term proportion that a certain outcome will occur in
situations where there is short-term uncertainty
o Probability deals with experiments that yield random short-term outcomes yet
reveal long-term predictability
o Essentially, the long-term proportion in which a certain outcome is observed is
the probability of that outcome
• Law of Large Numbers – as the number of repetitions of a probability experiment
increases, the proportion with which a certain outcome is observed gets closer to the
probability of the outcome.
• Experiment – a repetition of the same procedure in which the results are uncertain
• Sample Space (of a probability experiment) (S) – the collection of all possible outcomes
• Event (E) – any collection of outcomes from a probability experiment (ex: rolling a six-
sided die)
o Can have multiple outcomes
o Simple Events (ei) – events with one outcome (ex: flipping a two-sided coin)
• ex: consider the probability of having two children. Identify the outcomes of the
probability experiment, determine the sample space and define the event E = having one
boy
o e1 = boy + boy, e2 = boy + girl, e3 = girl + boy, e4 = girl + girl
o S = {(boy, boy), (boy, girl), (girl, boy), (girl, girl)}
o E = {(boy, girl), (girl, boy)}
Objective 2: Apply the Rules of Probabilities
• Rules of Probability:
o 1. The probability of any event P(E) – a positive number that must be greater than
or equal to 0 and less than or equal to 1
▪ 0 ≤ P(E) ≤ 1
▪ Probability of 1 = 100% chance of occurring
▪ Probability of 0 = 0% chance of occurring
o 2. The sum of the probabilities of all outcomes must equal 1.
o If the sample space S = {E1, E2,…En} then P(E1) + P(E2) +… P(En) = 1
• Probability Model – lists the possible outcomes of a probability experiment as well as
each outcome while satisfying both rules of probability
o If an event is impossible, the probability of the event is 0
o If an event is certain, the probability of the event is 1
o If an event has a low probability of occurring (less than 5%), it is an unusual
event
o ex: verify the probability model of M&Ms
find more resources at oneclass.com
find more resources at oneclass.com
▪ Rule 1 is satisfied, because all probabilities are between 0 and 1.
▪ Rule 2 is satisfied, because 0.12 + 0.15 + 0.12 + 0.23 + 0.23 + 0.15 = 1
Objective 3: Compute and Interpret Probabilities Using the Empirical Method
• Using the Empirical Method
• The probability of an event is approximately the number of times the event is observed
divided by the number of repetitions of the experiment
o P(E) – relative frequency of E
o P(E) =
• ex: build a probability model of the throws of the pig, calculate and interpret how many
throws of the pig would result in a side w/ dot and a leaning jowler
o Side w/ No Dot = 1344/3939 = 0.341, Side w/ Dot = 0.329, Razorback = 0.195,
Trotter = 0.093, Snouter = 0.035, Leaning Jowler = 0.008
o The probability a throw results in a “side w/ dot” is 0.329. In 1000 throws of the
pig, we would expect about 329 to land on a “side w/ dot”. The probability a
throw results in a “leaning Jowler” would be unusual. In 1000 throws of the pig,
we would expect to obtain the “leaning Jowler” about 8 times.
Objective 4: Compute and Interpret Probabilities Using the Classical Method
• Equally Likely Outcomes - a term used to describe an experiment when each simple
event has the same probability of occurring (ex: a coin flip)
o The classical method of computing probability requires equally likely outcomes
o This is usually assumed in most experiments
• Using the Classical Method
o If an experiment has ’n’ equally likely outcomes and if the number of ways the
event can occur is ‘m’, then…
▪ P(E) =
=
o So, if S is the sample space of this experiment,
▪ P(E) =
where N(E) is the number of outcomes in an event while N(S)
is the number of outcomes in the sample space
• ex: a M&M is randomly selected. What is the probability it is blue? Yellow?
o N(S) ~total amount of M&Ms in bag~ = 30
o P(yellow) = N(yellow)/N(S) = 6/30 = 0.2
o P(blue) = N(blue)/N(S) = 2/30 = 0.067
o Since P(yellow) = 6/30 and P(blue) = 2/30, selecting a yellow is three times as
likely as selecting a blue.
Objective 5: Recognize and Interpret Subjective Probabilities
• Subjective Probability (of an outcome) – a probability obtained on the basis of personal
judgement
o ex: an economist who predicts there is a 20% chance of recession next year
Section 5.2 – The Addition Rule and Complements
find more resources at oneclass.com
find more resources at oneclass.com
Objective 1: Use the Addition Rule for Disjoint Events
• Disjoint/Mutually Exclusive Events – events that have no outcomes in common
o ex: when rolling a pair of dice and (E = the event of rolling an even number while
F = the event of rolling an odd number) E and F are disjoint because there is no
outcome that can be in both events (no number can be both even and odd)
Objective 2: Use the General Addition Rule
• If E and F are disjoint (mutually exclusive) events, then P(E or F) = P(E) + P(F)
o ***or – can mean either or both
• The addition rule can be extended to more than two disjointed events
o If E, F, G, . . . each have no outcomes in common (they are pairwise disjoint),
then P(E or F or G) = P(E) + P(F) + P(G)
• If E and F are conjoined (overlapping) events, then P(E or F) = P(E) + P(F) – P(E and F)
o ***and – means both
o ex: if a pair of dice are thrown and E = the first die is a two while F = the sum of
the dice is less than or equal to 5. Find P(E or F) using the general addition rule.
▪ P(E) =
= 6/36 = 1/6
▪ P(E) =
= 10/36 = 5/18
▪ P(E and F) =
= 3/36 = 1/12
▪ P(E or F) = P(E) + P(F) - P(E and F) = 6/36 + 10/36 - 3/36 = 13/36
Objective 3: Compute the Probability of an Event Using the Complement Rule
• Complement of an Event (EC) – all the outcomes in a sample space that are not outcomes
in event E
o ex: if E = the first die is a two, then EC = the first die is anything but a two
• Important to note that E and EC are mutually exclusive
o P(E or EC) = P(E) + P(EC) = 1
o P(EC) = 1 – P(E)
• Important to note that E and EC are mutually exclusive
• ex: 31.6% of American households have a dog. What is the probability that a randomly
selected household does not have a dog?
o P(does not own a dog) = 1 - P(own a dog) = 1 - 31.6% = 0.684
• ex: roll two dice at random. What is the probability that at least one roll is a 1?
o E = (at least one roll is 1). Instead of finding P(E) we find the probability of the
opposite EC = neither roll is a 1
o P(E) = 1 - P(EC) = 1 - 25/36 = 11/36
Section 5.3 – Independence and the Multiplication Rule
Objective 1: Identify Independent Events
• Independent Events – when event E does not affect the probability of event F (ex:
conjoined events)
find more resources at oneclass.com
find more resources at oneclass.com
Document Summary
Objective 2: apply the rules of probabilities: rules of probability, 1. Trotter = 0. 093, snouter = 0. 035, leaning jowler = 0. 008: the probability a throw results in a side w/ dot is 0. 329. In 1000 throws of the pig, we would expect about 329 to land on a side w/ dot . The probability a throw results in a leaning jowler would be unusual. In 1000 throws of the pig, we would expect to obtain the leaning jowler about 8 times. Objective 5: recognize and interpret subjective probabilities: subjective probability (of an outcome) a probability obtained on the basis of personal judgement, ex: an economist who predicts there is a 20% chance of recession next year. Section 5. 2 the addition rule and complements. Objective 1: use the addition rule for disjoint events: disjoint/mutually exclusive events events that have no outcomes in common, ex: when rolling a pair of dice and (e = the event of rolling an even number while.
