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Lecture 6

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Department
Psychology
Course
PSYC 2002
Professor
Steven Carroll
Semester
Winter

Description
Lecture 6: probability People and probability - Imagine I flip a coin: • It comes up “heads” four times in a row • Is it more likely to come up “heads” or “tails” on the next throw? People and probability - How many things can happen on the fifth flip of the coin? (2) - How many of those things count as “I flipped heads”? (1) - How many of those things count as “I flipped tails”? (1) People and probability - What proportion of the total number of things that can happen would count as “heads”? • 1 / 2 = .5 - What proportion of the total number of things that can happen would count as “tails”? • 1 / 2 = .5 Gambler’s fallacy - The coin has no memory: It doesn’t know that it is due for a “tails” • The probability on any given throw (even after 100 throws of “heads”) reduces to: p(heads) = .5 and p(tails) = .5 - This same logic applies to dice games, roulette, slot machines, cards, and “strategically” picking lottery numbers: • Just because the game has tended to play out in one direction DOES NOT MEAN it is due to switch to another series of outcomes • These games have no memories Probability - What is it? Numberof possibleoutcomesclassifiedasA - P(A) = Thetotalnumberof possibleoutcomes Let’s play cards - There are 52 cards in a deck of cards - There are 4 suits - Each suit has 13 cards (ordinal distribution) • 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A - 4 suits X 13 cards per suit = 52 cards in a deck What is... - P(hearts) • = number of hearts / number of cards • = 13 / 52 = .25 - P(red card) • = number of reds / number of cards • = (13+13) / 52 = 26 / 52 = .5 - P(2) • = number of twos / number of cards • = 4 / 52 = .08 - P(2 of hearts) • 1 / 52 = .02 P(2 of hearts) - Another way to think of the p(2 of hearts) problem is this: - What is the combined p(2) AND p(heart) • P(2 ᴖ heart) - When you see the AND symbol “ᴖ” think “multiplication” - The formal name for this operation is the “intersection of sets” - P(2 ᴖ heart) • = p(2)p(heart) • = (4 / 52) (13 / 52) • = 52 / 2704 • = .02 What is... - P(2 OR jack) = ? • Another way to write this is p(2ᴗjack) • When you see the OR symbol “ᴗ” think “addition” • The formal name for this operation is the “union of sets) numberof thingsthatqualify - p(2ᴗjack) = numberof cards∈deck • = 4 twos + 4 jacks / 52 cards • = 8 / 52 = .15 Let’s play cards - Let’s play high-low: Stats edition! • I’ll pick a card. • Work out p(higher), p(lower), p(equal) • Then place an imaginary bet!  Or, if that isn’t palatable, just guess the next card category. • We’ll do this a couple of times... Let’s play cards - Did the bets always pay-off? What does that tell you about probability? - Is it likely that you’ll win the lottery? - P(win 6-49) = something you don’t have to know = - ************ 6-49 - P(win 6-49) = about 1 in 14 million chance - But is it strange thatANYONE wins the lottery? • Hint: sometimes they sell as many as 50 million tickets Let’s sample... - Somebody pick 5 cards out of the deck - Given the hand that has been removed from the deck, help my volunteer work out the remaining: - P(hearts) • = number of hearts / number of cards • We took 1 heart out, so now, it’s... • 12 / 47 = .2553 - P(red card) • = number of reds / number of cards • We took 2 red cards out, so now, it’s... • 24 / 47 = .5106 - P(2) • = number of twos / number of cards • No twos were removed, so now, it’s... • 4 / 47 = .0851 - P(2 of hearts) • No twos were removed, so now, it’s... • 1 / 47 = .0212 Consider the following - Now, we’ve talked about sampling and we’ve talked about the sampling bias - When you go to run an experiment you’re going to want to draw a representative sample from the population - Ideally, you want to draw people at random from the population: the goal is to draw a “RANDOM SAMPLE” - Random sample requires that each individual in the population has an equal chance of being selected. - Asecond requirement, necessary for MANY statistical formulae, states that the probabilities must stay constant from one selection to the next if more than one individual is selected. • Otherwise sampling isn’t random Think about that! - What is p(3 of clubs)? - What is p(4 of clubs) given I’ve already removed the three from the deck? - P(4 of clubs | no 3 of clubs in the deck) • “ | “ means “given” Violation - So this example has violated one of the rules of random sampling! - We call that “sampling without replacement” - So how could we perform “sampling with replacement” given the 3 and 4 of clubs example - Pretend I’m running an experiment and I draw a sample of students from the population of first-year university students. - There are 3138 students taking first-year psychology courses at Carleton. - What is p(student Bob Smith is chosen for the experiment)? - What is p(Jane Jones is chosen|
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