Warm Up Wed/Thurs.

1. Warm Up Wed/Thurs. • Do #17 p. 341 • You will want this on a piece of paper to KEEP for reference. • Have out HW for check: • Chapter 6.1-6.2 #’s 12,15,18,19,21,23,26,30

3. Homework • Finish Reading 6.2 • Do #’s 35 - 45 • Read and Take notes on Chapter 4.3 • p241-255 • do #’s 51,52,53,60,83

4. Final Ideas 6.1 • An event is any outcome or a set of outcomes of a random phenomenon. • An event is a subset of the sample space. • A probability model is a mathematical description of a random phenomenon. • Consists of two parts: a sample space, S and a way of assigning probabilities to events.

5. And remember we have ways to determine S and # of outcomes. • Tree Diagram – represent the first action, then draw “branches” to the next set of actions. • Multiplication Principle (of Counting) – do one task in “a” number of ways and another task in “b” number of ways, then both tasks can be done in “a” x “b” ways.

6. #11 • A) S = { germinates, fails to grow} • B) S = { 1,2,3,…..} depending on if you measure in days, weeks, months. • C) S = { A,B,C,D,F} • D) S = {HHHH, HHHM, HHMH, HMHH, MHHH, HHMM, HMHM, MMHH, MHMH, MHHM, HMMH, MMMH, MMHM, MHMM, HMMM, MMMM} • E) S = {4,3,2,1,0}

7. Discrete vs. Continuous • If the experiment is to throw a standard die and record the outcome, the sample space is: • S = {1, 2, 3, 4, 5, 6}, the set of possible outcomes. • Discrete sample space • Finite number of outcomes • On the other hand, if the experiment is to randomly pick a number between 0 and 1, then the sample space is to be S = [all #’s between 0 and 1], • Continuous sample space • Infinite number of outcomes

8. 6.12 • S = {0 – 24} • S = {0,1,2,3,...10,998, 10,999, 11,000} • S = {0,1,2,3,4,5,6,7,8,9,10,11,12} • S = {0, .01, .02, .03,…………} • S = {...negative grams, 0, positive grams…}

9. #14 • A) 2 x 2 = 4 • S = {hh, ht, th, tt} • B) 2 x 2 x 2 = 8 • S = {hhh, hht, hth, thh, tth, tht, htt, ttt} • C) 2 x 2 x 2 x 2 = 16 • {HHHH, HHHT, HHTH, HTHH, THHH, HHTT, HTHT, TTHH, THTH, THHT, HTTH, TTTH, TTHT, THTT, HTTT, TTTT}

10. I love probability  • 6.15 a) 10 X 10 X 10 X 10 10000 county license tags b) 10 X 9 X 8 X 7 5040 county license tags c) 104 + 103 + 102 + 101 11, 110 county license tags

11. 6.18 • Red card = 26 ways • Heart = 13 ways • Queen and a heart = 1 way • Queen or a heart = 13 ways for heart and 4 queens – 1 queen ♥ = 16 ways • Queen that is not a heart = 3 ways

12. Where do probabilities come from? • Probabilities may be given, often in the form of a table. For example, if an experiment has three possible outcomes: Apple, Banana, and Cherry, one might be given the following table:

13. Where do probabilities come from? • Probabilities may be historical, if it has rained during 1/3 of the days in June during the past, one may say that the probability of rain for a day in June is 1/3. • Probabilities may be theoretical, if a die is fair, since there are six possible outcomes; the probability of getting a 3 is 1/6.

14. Replacement v. Non-replacement • If you are selecting objects from a finite group of objects, whether you replace the object is very important. • If you do not replace, then the probability for each selection will change. • If we sample with replacement, each item is replaced in the population before the next draw; thus, a single object may occur several times in the sample. • If we sample without replacement, objects are not replaced in the population.

15. Problem: A glass jar contains 6 red, 5 green, 8 blue and 3 yellow marbles. If a single marble is chosen at random from the jar, what is the probability of choosing: • Probabilities: • P (RED) = • P (GREEN) = • P (BLUE) = • P (YELLOW) = What is the probability of getting 3 blue • With replacement • Without replacement

16. Conclusion: The outcomes in many experiments are not equally likely to occur. • You are more likely to choose a blue marble than any other color from the jar. • You are least likely to choose a yellow marble.

17. Probability Rules • 1. The probability P(A) of any event A satisfies . • 2. If S is the sample space in a probability model, then P(S) = 1. • 3. The complement of any event A is the event that A does not occur, written as . The complement rule states that • 4. If two events, A and B, have no outcomes in common (they are mutually exclusive or disjoint) then P(A or B) = P(A) + P(B). • 5. If two events A and B are independent (knowing that one occurs does not change the probability of the other occurring), then P(A and B) = P(A)*P(B). (This illustrates the multiplication rule for independent events.)

18. Probability is cool. • 6.19 • The probability of type AB blood would be 1-.49-.27-.20=.04 • P(O) + P (B) = .49 + .20 = .69 is the probability that a randomly chosen black American can donate to Maria. • 6.21 • The probability that the death was due to either cardio or cancer is .45+.22 = .67. • The probability that the death was due to some other cause is 1-.45-.22 = .33.

19. 6.23 a) The sum of the probabilities is 1. Probability Rule #2, all possible outcomes together must sum to 1. b) The probability that a randomly chosen first year student was not in the top 20% is 1-.41=.59 c) The probability that a randomly chosen first year student was in the top 40% is .41+ .23 =.64

20. Benford’s Law • D is defined as the 1st digit is less than 4, so P(D) = .301 + .176 + .125 = .602 • P( B U D) = (.067+.058+.051+.046) + (.602) = .824 • P(D)c = 1-.602 = .398 • P(C n D) = P ( odd and <4) = P (1) + P (3) = .301 + .125 = .426 • P(B n C) = P ( ≥6 and odd) = P (7) + P (9) = .058 + .046 = .104

21. 6.30 P(A)={the person chosen completed 4 yrs college} P(B)={the person chosen is 55 years old or older} a) P(A)= #people completing 4 yrs resident >25 yrs of age = 44845/175230 = .2559 or .256 b) P(B)=#people 55 yrs older resident >25 yrs of age = 56008/175230 = .3196 or .32 c) P(A and B)=#people 55 yrs older and 4 yrs of college resident >25 yrs of age = 10596/175230 = .0605 = .061

22. Homework • Finish Reading 6.2 • Do #’s 35 - 45 • Read and Take notes on Chapter 4.3 • p241-255 • do #’s 51,52,53,60,83