introduction to probability. take out a sheet of paper and a coin (if you don’t have one i will...
TRANSCRIPT
• Take out a sheet of paper and a coin (if you don’t have one I will give you one)
• Write your name on the sheet of paper.• When I leave the room:
• If the last digit of your ID # is odd, flip the coin 100 times, recording the heads and tails in order on the sheet
• If the last digit of your ID # is even, write down 100 heads or tails as if you were flipping a coin.
• I will leave for 5 minutes. When I come back I will guess who flipped the coins and who did not
How could I guess?
• Longest run of consecutive H or T
• What were you trying to do if you didn’t flip?– Make it look “random”
What is random?
• What are the odds that the first flip is a heads?– ½– Each outcome is equally likely
• The second flip?– ½
• So what are the odds that both are?– Four outcomes:
• HH, HT, TH, TT• so ¼ (each equally likely)
What is random?
• Odds the third flip is a heads?– ½
• Odds that all three are heads?– 8 outcomes– HHH, HHT, HTH, HTT, THH, THT, TTH, TTT– So, 1/8
• Odds the fourth flip is a heads?– ½
• All four?– 1/16
What is random?
• Odds that five in a row are heads?– 1/32
• Odds that six in a row?– 1/64
• How many sets of six are there in 100 flips?– 95 – (1-6, 2-7,…95-100)
We are bad at random
• Why didn’t “fake” flips have runs?– Didn’t “look random”
• What does that imply?– In the fake flips, the outcome of one flip is
dependent on past flips– Focused on short run, not long run
• Coins don’t have memories
• Expectations matter in the long run
Probability
• Definition:– Probability of an event is the number of times
that event can occur relative to the total number of times any event can occur.
Properties of probability
• The probability of an event is between 0 and 1
• If the events cannot occur simultaneously, then P(either) is the sum of the event probabilities.
Example
• What is the P(diamond) from a full deck of cards?– 0.25
• What is P(heart)?– 0.25
• What is p(red card)?– Diamond or heart– 0.25+0.25 = 0.50
Example
• What is probability of face card?– 3/13 = 0.23 (approx)
• What is probability of red card or face card?– Not the sum of the two (which would be .73).– How many cards are red or face?– 26 red cards, 6 black face cards– 32/52 = 0.62 (approx)
Properties of probability
• The probability of an event is between 0 and 1
• If the events cannot occur simultaneously, then P(either) is the sum of the event probabilities.
• Probability of that an event will not occur is 1-P(event)
Example
• What is probability that a card is neither a red card nor a face card?– 26 black cards, 20 of which aren’t face cards– = 20/52 – = 1-0.62– = 0.38
Properties of probability
• The probability of an event is between 0 and 1
• If the events cannot occur simultaneously, then P(either) is the sum of the event probabilities.
• Probability of that an event will not occur is 1-P(event)
• Sum of probabilities from all possible (mutually exclusive) is one.
Example
• Probability distribution for a single coin flip
Event Probability (P)
Heads ?
Tails ?
Total ?
Example
• Probability distribution for a single coin flip
Event Probability (P)
Heads 0.5
Tails 0.5
Total 1.0
Example
• Probability distribution for two coin flips
# of Heads Probability (P)
2 Heads ?
1 Heads ?
0 Heads ?
Total ?
Example
• Probability distribution for two coin flips
# of Heads Probability (P)
2 Heads .25
1 Heads ?
0 Heads ?
Total ?
Example
• Probability distribution for two coin flips
# of Heads Probability (P)
2 Heads .25
1 Heads ?
0 Heads .25
Total ?
Example
• Probability distribution for two coin flips
# of Heads Probability (P)
2 Heads .25
1 Heads .50
0 Heads .25
Total 1.0
What is random?
• What are the odds that the first flip is a heads?– ½– Each outcome is equally likely
• The second flip?– ½
• So what are the odds that both are?– Four outcomes:
• HH, HT, TH, TT• so ¼ (each equally likely)
Example
• Probability distribution for two coin flips
# of Heads Probability (P)
2 Heads .25
1 Heads .50
0 Heads .25
Total 1.0
Properties of Probability
• Independence: two events are independent if the chance of one event occurring is not affected by the outcome of the other event– Coin flips are independent
Independence
• Consecutive card draws would not be– P(first card is red) = 0.5– P(second card is red) = ?
• What if draw 1 is red?
Independence
• Consecutive card draws would not be– P(first card is red) = 0.5– P(second card is red) = ?– What if draw 1 is red?
• 25 red cards left out of 51• =25/51 • = 0.49
– What if draw 1 is black?• 26 red cards left out of 51• =26/51• = 0.51
Example
• I have a set of three cards– One is blue on both sides– One is pink on both sides– One is blue on one side pink on the other
• I will draw one without looking at the back side– What is the probability that the other side is Blue?– Pink?– Why?
Example
• Your turn!– Draw one card and tape it to the board without
looking at the other side
• Let’s see what we have
Summary of probability rules
• Addition rule for mutually exclusive events– P(outcome 1 or outcome 2) = P(outcome 1) + P(outcome 2)
• Complement rule– P(not outcome 1) = 1-P(outcome 1)
• Multiplication rule for independent outcomes– P(outcome 1 and outcome 2) = P(outcome 1) * P(outcome2)
• Multiplication rule for dependent outcomes– Much more complicated– Depends on the nature of the dependence
1. Get out one sheet of notebook paper 2. Put a heading on the paper Title: New Land, New Beginnings
Objective: Evaluate the causes and effects of the French Revolution Quiz-take out one sheet of paper