Chapter 6

Probability

What is it? -the likelihood of a specific outcome occurring

Why use it? -rather than constantly repeating experiments to make sure the same findings occur, scientists make probability statements

Criteria: -each specific event, must have an equally likely outcome *EX: Coin flip, the probability of getting heads must be equal to getting tails

-the probability (p) assigned to each specific event must be equal to or greater than zero OR equal to or less than 1.00. *p=0 means the event can’t occur & p=1.00 means it must occur *no p-value can fall outside these limits

-the sum of the probabilities assigned to all the specific events must equal 1.00. *EX: Coin flip, heads p=.50 & tails p=.50

Probabilities Expressed as Symbols: -decimal: p=.50 *ie. An event will occur 50 times out of 100 -percentage: p=50% *ie. An event will occur 50% of the time -fraction: p=1/2 *ie. An event will occur 1 out of every 2 times -using “less than sign”: p < .05 *means the probability is less than .05

Two types:1. Expected Relative Frequency: the outcome you expect to get if you repeat an event a large number of times -EX: coin flip, expect to get heads 1 out of 2 times (or ½)

2. Subjective Interpretation of Probability: how certain you are that a particular thing will happen -EX: “95% chance that my favorite restaurant is open tonight”

Probability

ProbabilityClassical Model: calculate BEFORE observations are made • assumes that the alternative events are mutually exclusive -ie. ONLY ONE CAN be the observed outcome• are collectively exhaustive -ie. one of these outcomes MUST result• have equal likelihood

Calculation:Probability=Possible Successful Outcomes/All Possible Outcomes -EX: Probability of getting a 3 or lower when rolling dice *Possible Successful Outcomes=3 (Could roll a 1, 2 0R 3) *All Possible Outcomes= 6 (Could roll a 1, 2, 3, 4, 5, 6) *p=3/6=.50

Calculating the probability of multiple outcomes: -EX: Probability of rolling a 3 two times in a row? *p=(1/6)(1/6) = .17 x .17 = .03

Probability

Empirical Model: calculate AFTER observations have been made•used when events are NOT equally likely -EX: a loaded coin weighted to land on heads (so tails is NOT equally likely)•events are mutually exclusive -ie. one coin can’t land on heads & tails at the same time•past observations are used to calculate future probability -calculation assumes there have been a lot of past observations made

Calculation: p = number of past observations for an outcome / total observations -EX: What is the probability that on this 3rd down the Chargers will convert? -converted four 3rd downs so far this game (past observations) -had 16 3rd down opportunities so far this game (total observations) -p=4/16 or .25 or 25%

Gambler’s Fallacy: assuming that after a losing streak, the probability must swing the other way & a win will surely occur. -WRONG! Events are independent of each other *ie. The outcome of the previous event has no effect on the next event

Probability versus Odds: -probability states the likelihood events will occur -odds state the likelihood event will not occur *EX: What is the likelihood of rolling a die & getting any number? -Probability=1/6 -Odds= 5 to 1 (because there are 5 sides that won’t come & 1 that will) -To calculate from probability to odds: odds=p/(1-p) *EX: p=.20 odds=.20/(1-.20)=.20/.80 or 1/4 read/written as “4 to 1 against” OR *EX: p=.80 odds=.80/(1-.80)=.80/.20 or 4/1 read/written as “4 to 1 in favor”

Probability

Gambler’s Fallacy

ProbabilityNormal Distribution, Z-scores &Probability: -the normal curve allows us to make statements about probability -the percentages we calculated with z-scores can be used as probability ie. Calculations about the area between, above or below z-scores -convert percentages to probability: p= percentage/100

EX: On a normal distribution with M=25 & SD=2.42, what is the probability that any single score will fall above a raw score of 27? -calculate the z-score: z=27-25/2.42=.83

-look up percentage: 29.67% -subtract it from 50 50%-29.67%=20.33% -convert to probability p=20.33/100=.20 p=.20