CAS MA 113 Lecture Notes - Lecture 2: Binge Drinking, Sample Space, Pepperoni
30 Apr 2018
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Chapter 5:
Probability
•Probability - a measure of the likelihood of a random phenomenon/chance behavior occurring.
•Describes the long-term proportion with which a certain outcome will occur in situations with
short-term uncertainty
•Essentially, a way to quantify how likely an outcome is.
•Probability is involved in experiments that yield random outcomes - short-term results - yet revealing
long-term predictability
•The long-term proportion in which a certain outcome is observed is the probability of that outcome
•The Law of Large Numbers - as the number of reputations of a probability experiment increases, the
proportion with which a certain outcome is observed gets closer to the probability of the outcome
•Random Experiment - an act or process of observation that leads to a single outcome that cannot be
predicted with certainty
•Sample Point - the most basic outcome of a random experiment
•Sample Space - the collection of all the sample points
•ex: consider the probability experiment of having three children. Identify the outcomes of the
probability experiment and determine the sample space. Define the event E = ‘have one boy’
•Sample space: BBB BBG BGB GBB BGG GBG GGB GGG
•Sample space: BBB BBG BGB GBB BGG GBG GGB GGG
•E={(B,G,G), (G,B,G), (G,G,B)} = 3/8 probability
•Probability Model - lists the possible outcomes of a probability experiment as well as each
outcome’s probability
•This model must satisfy both rules of probability
•Rules of Probability
•The probability of any event [P(e)] - must be greater than or equal to 0 and less than or equal to
1 (between 0 and 1, inclusive)
•0 ≤ P(E) ≤ 1
•The sum of the probabilities of all outcomes must equal 1.
•if the sample space S = {e1, e2,…en} then P(e1) + P(e2) +… P(en) = 1
•If an event is impossible, the probability of the event is 0
•If an event is certain, the probability of the event is 1
•Unusual Event - an event that has a low probability of occurring (usually said if an event has a
probability less than 5%)
Computing Probability
•Using the Empirical Approach (approximate)
•The probability of an event is approximately the number of times the event is observed divided by
the number of repetitions of the experiment
•P(E) - relative frequency of E
•P(E) = (frequency of E)/(number of trials of the experiment)
•The classical method of computing probabilities requires equally likely outcomes
•Equally Likely Outcomes - a term used to describe an experiment when each simple event has the
same probability of occurring (ex: a coin flip)
•This is usually assumed in most experiments
•Using the Classical Method
•If an experiment has ’n’ equally likely outcomes and if the number of ways the event can occur is
‘m’, then -
•P(E) = (number of ways E can occur)/(number of possible outcomes) = m/n
•So, if S is the sample space of this experiment,
•P(E) = N(E)/N(S)
•where N(E) is the number of outcomes in an event while N(S) is the number of outcomes in
the sample space
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•Disjoint/Mutually Exclusive - two events are disjoint if they have no outcomes in common
•(ex: rolling a dice, 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 experiment outcome that can be in both events (no
number can be both even and odd))
•Addition Rule for Disjoint Events
•If E and F are disjointed then P(E or F) = P(E) + P(F)
•***or - can mean either or both
•Venn Diagrams - this diagram uses pictures to represent events as circles enclosed in a rectangle
•The rectangle represents the sample space while each circle represents and vent
•The General Addition Rule ~ Addition Rule for Conjoined Events - If E and F are overlapping
then P(E or F) = P(E) + P(F) - P(E and F)
•ex: suppose a pair of dice are thrown. Let E = the first die is a two. Let F = the sum of the dice is
less than or equal to 5. Find P(E or F) using the general addition rule.
•answer:
•P(E) = [N(E)]/[N(S)] = 6/36 = 1/6
•P(F) = [N(F)]/[N(S)] = 10/36 = 5/18
•P(E and F) = [N(E and F)]/[N(S)] = 3/36 = 1/12
•P(E or F) = P(E) + P(F) - P(E and F) = 6/36 + 10/36 - 3/36 = 13/36
•Complement of an event (EC) - all the outcomes in a sample space that are not outcomes in event E
•ex: if E = the first die is a two then EC = the first die is anything but a two
•Naturally E and EC are mutually exclusive
•P(E or EC) = P(E) + P(EC)
•Every simple event is either in E or it isn’t, hence making it EC
•P(E or EC) = 1
•Complement Rule - if E represents any event and EC represents the complement of E then P(EC) = 1
= P(E)
•ex: 31.6% of American households have a dog. What is the probability that a randomly selected
household does not have a dog?
•answer: 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 at least one roll is a 1?
•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
•P(E) = 1 - P(EC) = 1 - 25/36 = 11/36
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•ex: a table shows the favorite pizza topping for a sample of students. One of these students is selected
at random. Find the probability that the student is male or prefers any meat.
•answer 1:
•E = (student is male or likes meat) ~ (student is male and anyone who likes pepperoni/sausage)
•EC = (student is not male or does not like meat) = (student is female and likes cheese)
•E = 1 - EC = 1 - 2/24 = 22/24 = 0.917
•E and F are independent - if the occurrence of event E in a probability experiment does not affect
the probability of event F
•ex: rolling two die - the first number has no effect on the second number
•E and F are dependent - if the occurrence of event E in a probability experiment does affect the
probability of event F
•ex: choosing playing cards - the first card will effect the possibility of the second card
Multiplication Rules
•Multiplication Rule for Independent Events
•If E and F are independent events then P(E and F) = P(E) x P(F)
•ex: the probability that a woman will survive the year is 99.186%. What is the probability that
two randomly selected women will survive the year?
•answer: the survival of the first female is independent of the second female - P(survive) =
0.99186
•P(first and second survives) = P(first survives) x P(second survives) = (0.99186)(0.99186) =
0.9838
•Multiplication Rule for ’n’ Independent Events
•If E1, E2, E3… and En are independent events then
•P(E1 and E2 and E3 and En) = P(E1) x P(E2) x P(E3) x P(En)
•1. ex: the probability that a woman will survive the year is 99.186%. What is the probability that
four randomly selected women will survive the year?
•answer: the survival of the first female is independent of the second female - P(survive) =
0.99186
•P(first, second, third and fourth survives) = (0.99186)(0.99186)(0.99186)(0.99186) = 0.9678
•2. ex: roll seven dice and find the probability that at least one die roll is a 1.
•E = (at least one roll is not a 1) and EC = (no roll is a 1)
•EC = (1st die is not a 1) and (2nd die is not a 1) and (3rd die is not a 1)…and (6th die is not a 1)
•P(EC) = P(1st die is not a 1) x P(2nd die is not a 1) x P(3rd die is not a 1)…x P(6th die is not a
1)
•= 5/6 x 5/6 x 5/6…x 5/6 = (5/6)7 = 0.2791
•P(E) = 1 - P(EC) = 1 - 0.2791 = 0.7209
•3. ex: 44% of college students engage in binge drinking. If five students are randomly selected
what is the probability that at least one of them have engaged in binge drinking?
•E = (at least one student drinks) and EC = (no student drinks)
•EC = (1st student does not drink) and (2nd student does not drink)…and (5th student does not
drink)
•P(EC) = P(1st student does not drink) x P(2nd student does not drink)…x P(5th student does not
drink)
•= 0.56 x 0.56 x 0.56 x 0.56 x 0.56 = 0.055
Cheese
Pepperoni
Sausage
TOTAL
Male
8
5
2
15
Female
2
4
3
9
TOTAL
10
9
5
24
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find more resources at oneclass.com
Document Summary
Identify the outcomes of the probability experiment and determine the sample space. Let e = the first die is a two. Let f = the sum of the dice is less than or equal to 5. = p(e: ex: 31. 6% of american households have a dog. What is the probability that a randomly selected household does not have a dog: answer: 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 at least one roll is a 1: e = (at least one roll is 1). One of these students is selected at random. Find the probability that the student is male or prefers any meat. Multiplication rules: multiplication rule for independent events, if e and f are independent events then p(e and f) = p(e) x p(f, ex: the probability that a woman will survive the year is 99. 186%.
