How much sleep did you get last night?

1. How much sleep did you get last night? • <6 • 6 • 7 • 8 • 9 • >9

2. Upcoming In Class • Thursday Quiz 2, covers HW3 of the material learned in class. (2/7) • Sunday HW4 (2/10) • Part 2 of the data project due next Tuesday (2/12)

3. Chapter 14 From Randomness to Probability

4. Dealing with Random Phenomena • A random phenomenon is a situation in which we know what outcomes could happen, but we don’t know which particular outcome did or will happen. • In general, each occasion upon which we observe a random phenomenon is called a trial. • At each trial, we note the value of the random phenomenon, and call it an outcome. • When we combine outcomes, the resulting combination is an event. • The collection of all possible outcomes is called the sample space.

5. Which of the following is the sample space for recording the order of heads and tails when tossing 3 coins? • {HHT, HTT, HTH, TTT} • {HHH,HHT,HTH,HHT, HTT, THT,TTH, THH, TTT} • {H,T, HH, HT, TH TT} • {H,T}

6. A family has 5 children. What is the sample space for the number of boys in the family? • {1,2,3,4,5} • {0,1,2,3,4,5} • {0,1,2,3,4} • {1,2,3,4}

7. During a Basketball game a player Takes a Shot. What is the sample space for the Number of Points? • {0, 1, 2, 3} • {-1, 1, 2, 3} • {1, 2, 3} • {0, 1, 2}

8. From Random Events to Modeling Probability • When probability was first studied, a group of 17th century French mathematicians looked at games of chance in which all the possible outcomes were equally likely. • It’s equally likely to get any one of six outcomes from the roll of a fair die. • It’s equally likely to get heads or tails from the toss of a fair coin. • However, keep in mind that events are not always equally likely. • A skilled basketball player has a better than 50-50 chance of making a free throw.

9. Modeling Probability with Relative Frequency • The probability of an event is the number of outcomes in the event divided by the total number of possible outcomes. P(A) = # of outcomes in A # of possible outcomes

10. The Law of Large Numbers First a definition . . . • When thinking about what happens with combinations of outcomes, things are simplified if the individual trials are independent. • Roughly speaking, this means that the outcome of one trial doesn’t influence or change the outcome of another. • For example, coin flips are independent.

11. The Law of Large Numbers (cont.) • The Law of Large Numbers (LLN) says that the long-run relative frequency of repeated independent events gets closer and closer to a single value. • We call the single value the probability of the event. • Because this definition is based on repeatedly observing the event’s outcome, this definition of probability is often called empirical probability.

12. Event Simulator • Coin Toss LLN Experiment (Redline) • http://socr.ucla.edu/htmls/SOCR_Experiments.html

13. Personal Probability • In everyday speech, when we express a degree of uncertainty without basing it on long-run relative frequencies, we are stating subjective or personal probabilities. • Personal probabilities don’t display the kind of consistency that we will need probabilities to have, so we’ll stick with formally defined probabilities.

14. Formal Probability • Two requirements for a probability: • A probability is a number between 0 and 1. • For any event A, 0 ≤ P(A) ≤ 1.

15. Formal Probability (cont.) • Probability Assignment Rule: • The probability of the set of all possible outcomes of a trial must be 1. • P(S) = 1 (S represents the set of all possible outcomes.)

16. Enter question text... • A,B,C,D,E • A,B,C • A,B,D,E • A,B,C,D

17. The 68-95-99.7 Rule (cont.) • The following shows what the 68-95-99.7 Rule tells us:

18. Formal Probability (cont.) • Complement Rule: • The set of outcomes that are not in the event A is called the complement of A, denoted AC. • The probability of an event occurring is 1 minus the probability that it doesn’t occur: P(A) = 1 – P(AC)

19. Formal Probability (cont.) • Addition Rule: • Events that have no outcomes in common (and, thus, cannot occur together) are called disjoint (or mutually exclusive).

20. Which of the following events are disjoint? • Being black and female • Being republican and female • Being female and an athlete • Being female and having a Y chromosome

21. Formal Probability (cont.) • Addition Rule (cont.): • For two disjoint events A and B, the probability that one or the other occurs is the sum of the probabilities of the two events. • P(AorB) = P(A) + P(B),provided that A and B are disjoint.

22. Enter question text... • =513/1121 • =511/1121 • =62/1121 • =35/1121

23. Enter question text... • =513/1121 • =511/1121 • =(62+35)/1121 • =35/62

24. Formal Probability • Multiplication Rule (cont.): • For two independent events A and B, the probability that both A and B occur is the product of the probabilities of the two events. • P(AandB) = P(A) x P(B), provided that A and B are independent.

25. Formal Probability - Notation Notation alert: • In this text we use the notation P(A or B) and P(A and B). • In other situations, you might see the following: • P(AB) instead of P(A or B) • P(AB) instead of P(A and B)

26. Enter question text... • 1/3 • 3 • 1/6 + 1/6 +1/6 • 1/6 * 1/6 * 1/6

27. Enter question text... • 1/2 • 2 • 1/2 + 1/2 +1/2 • 1/2 * 1/2 * 1/2

28. Enter question text... • 2/6*2/6*2/6 • 2/6+2/6+2/6 • 1- 2/6+2/6+2/6 • 4/6 * 4/6 * 4/6

29. Enter question text... • 1/6*1/6*1/6 • 1-1/6*1/6*1/6 • 1-5/6*5/6*5/6 • 5/6 * 5/6 * 5/6

30. Enter question text... • 1/6*1/6*1/6 • 1-1/6*1/6*1/6 • 1-5/6*5/6*5/6 • 5/6 * 5/6 * 5/6

31. What Can Go Wrong? (cont.) • Don’t multiply probabilities of events if they’re not independent. • The multiplication of probabilities of events that are not independent is one of the most common errors people make in dealing with probabilities. • Don’t confuse disjoint and independent—disjoint events can’t be independent.

32. Upcoming In Class • Thursday Quiz 2, covers HW2 and HW3 of the material learned in class. (9/12) • Sunday HW4 (9/16) • Part 2 of the data project due next Monday (9/17)