Contemporary Math Exam 2

Appeal to Popularity (bandwagon)

people think because something is popular then it is good or correct

ex.) Premise: Nike is the best-selling brand of athletic footwear

Conclusion: Nike shoes are better than other brands for sports performance

False Cause

believing that an event caused another because it came before that event

ex.) Premise: yesterday the groundhog saw its shadow

premise: today the weather forecast predicts six more weeks of winter

conclusion: the groundhog seeing its shadow delayed the arrival of spring weather

Appeal to Ignorance

lack of knowledge about the truth to conclude the opposite

ex.) “no proof… no evidence..”

Hasty Generalization

takes a few examples and make a broad generalization

ex.) premise: the governor was found guilty of tax evasion

premise: three senior government officials were convicted of embezzlement

conclusion: all politicians are corrupt

Limited Choice (false dilemma)

presenting only two options when more exist

ex.) “you can either join the gym or be unhealthy”

Appeal to Emotion

evoke an emotional response as a tool of persuasion

ex.) “if you don’t invest in this opportunity now, you’ll regret it forever.”

Personal Attack

attacking the character, motives, etc. of an individual or group

ex.) “you can’t trust his opinion on fitness, he’s not even healthy himself.”

Circular Reasoning

premise and conclusion both essentially say the same thing

ex.) premise: I have a right to a late curfew

conclusion: you should allow me to come home at 11 PM

Diversion (Red Herring)

divert attention from main issue by focusing on another issue, usually irrelevant

ex.) friend: “i feel like I’m always the one making plans.”

other friend: “but we always have fun when we hang out!”

Strawman

distorting someone’s words/beliefs to create an argument, usually extreme

ex.) premise: the president is advocating for one trillion-dollar stimulus bill

premise: his opponents say that the president is in favor of uncontrolled government spending which they believe will lead to inflation

conclusion: congress should not pass the stimulus bill

Negation

opposite of a proposition (use the word NOT)

Conjunction

and statements, only true if p & q are true

Disjunction

or statements, true unless p & q are both false

Conditional (original statement)

if p, then q

Converse (change order)

if q, then p

Inverse (INsert Not)

if not p, then not q

Contrapositive (Both)

if not q, then not p

Subset

all members of A are also members of B, A is a subset of B

Disjoint

two sets have no members in common

Overlapping

two sets that share SOME of the same members

Inductive Arguments

specific premises → general conclusion (premises support conclusion, canNOT guarantee to be true)

ex.) premise: falcons can fly, hummingbirds can fly

conclusion: all birds can fly (false)

Deductive Arguments

general premises → specific conclusions

ex.) premise: all doctors are intelligent

premise: Dr. Jones is a doctor

conclusion: Dr. Jones is intelligent

Affirming the Hypothesis

p is true, q is true VALID

Affirming the Conclusion

q is true, p is true INVALID

Denying the Hypothesis

p is not true, q is not true INVALID

Denying the Conclusion

q is not true, p is not true VALID

Standard Deck of Cards

52 cards: two colors (26 cards each), four suits (13 cards each)

Theoretical Method for Equally Likely Outcomes (what should happen)

P(A) = # of ways A can occur / total # of outcomes

Relative Frequency (empirical method, data that has been collected)

P(A) = # of times A occurred / total # of observations

Probability Distribution

list all possible outcomes in a table and determine the probability of each event, sum of all probabilities equal 1

Odds For

odds for event A are P(A) / P(not A)

Odds Against/On

odds against/on event A are P(not A) / P(A)

And Probability: Independent Events

P (A and B) = P(A) x P (B)

ex.) flipping a coin, rolling dice

And Probability: Dependent Events

P (A and B) = P(A) x P(B given A)

ex.) drawing cars without replacement

Either / Or Probabilities: Mutually Exclusive Events

P(A or B) = P(A) + P(B)

ex.) disjoint, even and odd numbers

Either / Or Probabilities: Overlapping Events

P(A or B) = P(A) + P(B) - P(A and B)

ex.) outcome of picking a queen or a club

At Least Once Rule (for independent events)

P (at least one event A in n trials) = 1 - [P(not A)]ⁿ

P (at least one) = 1 - P (none)

^ the opposite of it not happening

Expected Value

expected value = (event 1 value) x (event 1 probability) + (event 2 value) x (event 2 probability)

Permutation of a Selection of Items

_nP_r = n! / (n - r)!

• order or arrangement matters

“different, distinct”

Combinations

_nC_r = n! / (n-r)! x r!

• order or arrangement does NOT matter

Fundamental Counting Principle

total number of ways a compound event may occur is n1 x n2 x n3 (n1 = # of ways the first event may occur etc.)