PSY201: Chapter 6 Flashcards
Introduction to Probability
method for measuring + quantifying likelihood of obtaining specific sample from specific pop
fraction/proportion
probability of specific outcome determined by ratio comparing frequency of occurrence for outcome relative to total # of possible outcomes
Introduction to Probability
possible events of interest/# all possible events
Scrabble tiles with the letters A, B, C, & D in a bag, probability of selecting B is:
fraction: p(Btile)=1⁄4
proportion: p(Btile)=1in4 or 0.25
Probability of independent event
occurrence of one outcome has no effect on likelihood of occurrence of next event
probability of “tails” is always 0.5, even if, previous flip was a “tails”, previous 2/5 flips were “tails”
Gambler’s fallacy
Probability of dependent event
Occurrence of 1 outcome affects probability of another event’s occurrence
sampling without replacement
Specific Addition Rule
either/rule for mutually exclusive events
if A or B (or C etc.) will occur
Events must be independent of each other
cannot occur at same time/influence each other
P(A/B) = P(A) + P(B)
General Addition Rule
(still either/rule) for events that are not mutually exclusive
events are not mutually exclusive as it is possible for a card to be both a King + a Spade
P(A/B) = P(A) + P(B) – P(A + B)
Specific Multiplication rule: Probability of sequence of events
Only valid for independent events: 2 events independent if occurrence of one doesn’t change probability of other occurring
same probability because they are independent
Multiplication rule: Probability of sequence of events
Probability of them both occurring is product of probabilities of each occurring
P(A + B) = P(A) x P(B)
Conditional Probability: For non- independent events
probability that event B occurs given that event A has occurred
denoted by symbol P(B | A): probability of B given A
Conditional Probability: For non- independent events
What is probability of pulling a black card, not replacing it + then pulling a red card from the deck
probability of A x probability of B, given that A has already occurred = p(A) × p(B|A)
Probability
necessary to assume random sampling: has 2 requirement
Probability
(1) Each indiv in pop has equal chance of being selected
(2) If >1 indiv is selected, probability must stay constant for all selections - random selection with replacement
Probability
scores in pop variable ⇒ impossible to predict with perfect accuracy exactly which score or scores will be obtained when you take a sample from the population.
researchers rely on probability to determine relative likelihood for specific samples
possible to determine which outcomes have high probability + which have low probability
Probability and the normal distribution
normal distribution useful because many naturally occurring phenomena are distributed in ‘normal’ fashion
pop height/weight/IQ
Probability and the normal distribution
Almost all will fall between -3 + +3 SDs from mean
Can compare distributions with each other
Certain % will fall between different points
The Theoretical Normal Curve
General relationships:
±1 s = about 68.26%
±2 s = about 95.44%
±3 s = about 99.72%
de Moivre and the normal curve
Fled France for England late 17th century
Good friend of Isaac Newton
Expert on chance
Credited for discovering concept of normal curve
first to compute areas under curve as 1, 2, & 3 SDs
Carl Gauss
normal curve often called Gaussian distribution, after German mathematician Carl Friedrich Gauss, who discovered many of its properties
Properties of the Normal Curve
Theoretical construction
Bell Curve/Gaussian Curve
Perfectly symmetrical normal distribution
mean of a distribution is midpoint of curve
Properties of the Normal Curve
tails of the curve are infinite
Mean of curve = median = mode
“area under the curve” is measured in standard deviations from mean
Probability and the Normal Distribution
vertical line is drawn through a normal distribution, the following occur:
- exact location of line can be specified by z-score
- line divides distribution into two sections: larger section (body) + smaller section (tail)
unit normal table
lists several different proportions corresponding to each z-score location
Column A: z-score values
columns B + C: proportions in body + tail, respectively.
Column D: proportion betw mean + z-score location
table values can also be used to determine probabilities
Probability and the Normal Distribution
first transform X into z- score ⇒ look up z-score in table + find appropriate proportion/probability
Probability and the Normal Distribution
To find X for particular proportion: look up proportion in table + find corresponding z-score ⇒ transform into X
Probability and the Binomial Distribution
formed by series of observations for which there are exactly 2 possible outcomes (heads + tails)
Probability and the Binomial Distribution
A + B
p(A) = p + p(B) = q
distribution shows probability for each X (# of occurrences of A in series of n)
Probability and the Binomial Distribution
When pn + qn are both > 10, binomial distribution closely approximated by normal distribution
mean: μ = pn + SD: σ = √npq
z-score can be computed for each X + unit normal table can be used to determine probabilities for specific outcome
Probability and Inferential Statistics
Probability establishes link betw samples + pops
For any known population it is possible to determine probability of obtaining any specific sample
Probability and Inferential Statistics
general goal of inferential statistics: use info from sample to reach general conclusion (inference) about unknown pop
Typically researcher begins with sample
Probability and Inferential Statistics
sample has high probability of being obtained from specific pop ⇒ researcher can conclude that sample is likely to come from that pop
sample has very low probability of being obtained from specific pop ⇒ reasonable for researcher to conclude specific population is probably not source for sample
