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– 0.25+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)

? ?

? ?

Total ?

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

Probability (P)

Total

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

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 Pink

Pink

Blue, Pink Blue

Pink

Blue, Blue Blue

Blue

1/3

1/3

1/3

1/2

1/2

1/2

1/2

1/2

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