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PSYC 2002
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Steven Carroll
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Lecture 6

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Psychology

PSYC 2002

Steven Carroll

Winter

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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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