Introduction to probability

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 – = 0.50

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 –= –= 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 EventProbability (P) ?? ?? Total?

Example Probability distribution for a single coin flip EventProbability (P) Heads? Tails? Total?

Example Probability distribution for a single coin flip EventProbability (P) Heads0.5 Tails0.5 Total1.0

Example Probability distribution for two coin flips Probability (P) Total

Example Probability distribution for two coin flips # of HeadsProbability (P) 2 Heads? 1 Heads? 0 Heads? Total?

Example Probability distribution for two coin flips # of HeadsProbability (P) 2 Heads.25 1 Heads? 0 Heads? Total?

Example Probability distribution for two coin flips # of HeadsProbability (P) 2 Heads.25 1 Heads? 0 Heads.25 Total?

Example Probability distribution for two coin flips # of HeadsProbability (P) 2 Heads.25 1 Heads.50 0 Heads.25 Total1.0

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 HeadsProbability (P) 2 Heads.25 1 Heads.50 0 Heads.25 Total1.0

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

Example Your turn! –Draw one card and tape it to the board without looking at the other side Let’s see what we have

Example Are they 50/50?

Example Are they 50/50? Why not?

Pink,Pink Blue,PinkBlue Pink Blue,Blue 1/3 1/2

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