Introduction to probability

1. Introduction to probability

2. 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

3. 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”

4. 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)

5. 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

6. 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)

7. 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

8. 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.

9. 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.

10. 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

11. 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)

12. 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)

13. 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

14. 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.

15. Example • Probability distribution for a single coin flip

16. Example • Probability distribution for a single coin flip

17. Example • Probability distribution for a single coin flip

18. Example • Probability distribution for two coin flips

19. Example • Probability distribution for two coin flips

20. Example • Probability distribution for two coin flips

21. Example • Probability distribution for two coin flips

22. Example • Probability distribution for two coin flips

23. 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)

24. Example • Probability distribution for two coin flips

25. 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

26. 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?

27. 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

28. 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?

29. Example • Your turn! • Draw one card and tape it to the board without looking at the other side

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

31. Example • Are they 50/50?

32. Example • Are they 50/50? • Why not?

33. 1/2 1/2 1/3 1/3 1/2 1/2 1/3 1/2 1/2

34. 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