1 / 20

POD #49 11/29/2011 2003B #2

POD #49 11/29/2011 2003B #2. What is the marginal distribution of age? What proportion of the sample made over $50,000?. Stats: Modeling the World. Chapter 14 From Randomness to Probability. Sample Space. The set of all possible outcomes Try it!. Albert/Barbara Albert/Carl

cramon
Download Presentation

POD #49 11/29/2011 2003B #2

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. POD #49 11/29/2011 2003B #2 What is the marginal distribution of age? What proportion of the sample made over $50,000?

  2. Stats: Modeling the World Chapter 14 From Randomness to Probability

  3. Sample Space The set of all possible outcomes Try it! Albert/Barbara Albert/Carl Albert/Dianna Barbara/Carl Barbara/Dianna Carl/Dianna AND VICE VERSA!!!! P(Both Girls) = 2/12

  4. Basic Probability Rules • For any event A, 0 ≤ P(A) ≤ 1. • The sum of the probabilities for all outcomes in the sample space must equal 1. P(S) = 1

  5. Try it! Is this a possible sample space? Why or why not? a) {0.25, 0.25, 0.25, 0.25} b) {0.1, 0.2, 0.3, 0.4} c) {0.2, 0.3, 0.4, 0.5} d) {0, 0, 1, 0} e) {0.1, 0.2, 1.2, -1.5} Yes Yes No Yes No

  6. Addition Rule P(AorB) = P(A) + P(B) provided that A and B are disjoint. Disjoint (Mutually Exclusive) The two events, A and B, cannot occur at the same time.

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

  8. Try it!

  9. Multiplication Rule P(AandB) = P(A) x P(B) provided that A and B are independent. Independent Events Knowledge of event A does not affect the probability of event B

  10. Try it!

  11. Example The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&Ms, red another 20%, and orange, blue, and green each made up 10%. The rest were brown. a) If you pick an M&M at random, what is the probability that: - it is brown? - it is yellow or orange? - it is not green? - it is striped? 30% 20%+10% = 30% 100% - 10% = 90% 0%

  12. Example The Masterfoods company says that before the introduction of purple, yellow candies made up 20% of their plain M&Ms, red another 20%, and orange, blue, and green each made up 10%. The rest were brown. b) If you pick three M&Ms in a row, what is the probability that: - they are all brown? - the third one is the first one that’s red? - none are yellow? - at least one is green? (.3)(.3)(.3) = 0.027 (.8)(.8)(.2) = 0.128 (.8)(.8)(.8) = 0.512 1-No Greens = 1 - (.9)(.9)(.9) = 0.271

  13. Example A consumer organization estimates that over a 1-year period 17% of cars will need to be repaired once, 7% will need repairs twice, and 4% will require three or more repairs. a) What is the probability that a car chosen at random will need: 1. no repairs? 2. no more than one repair? 3. some repairs? P(0) = 1 – 0.28 = 0.72 P(0 or 1) = 0.72 + 0.17 = 0.89 P(1 or more) = 1 – P(none) = 0.28

  14. Example A consumer organization estimates that over a 1-year period 17% of cars will need to be repaired once, 7% will need repairs twice, and 4% will require three or more repairs. b) If you own two cars, what is the probability that: 1. neither will need repair? 2. both will need repair? 3. at least one will need repair? P(none and none) = (0.72)(0.72) = 0.5184 P(some and some) = (0.28)(0.28) = 0.0784 1 - P(none) = 1 – 0.5184 = 0.4816

  15. Example A certain bowler can bowl a strike 70% of the time. What is the probability that she a) goes three consecutive frames without a strike? b) makes her first strike in the third frame? c) has at least one strike in the first three frames d) bowls a perfect game (12 consecutive strikes) P(no/no/no) = (0.3)(0.3)(0.3) = 0.027 P(no/no/yes) = (0.3)(0.3)(0.7) = 0.063 1 - P(no/no/no) = 1 – 0.027 = 0.973 P(12 strikes) = (0.7)^12 = 0.0138

  16. Example Suppose that in your city 37% of voters are registered Democrats, 29% Republicans, and 11% other parties. Voters not aligned with any party are termed “independent”. You are conducting a poll by calling registered voters at random. In your first three calls, what is the probability you talk to: a. All Republicans? b. No Democrats? c. At least one Independent? P(RRR) = (0.29)(0.29)(0.29) = 0.024 P(ND/ND/ND) = (0.63)(0.63)(0.63) = 0.250 1 – P(no Indy)= 1 - (0.89)(0.89)(0.89) = 0.295

  17. P(40 or 100) = 3/12 = 0.25 P(10/10) = (0.5)(0.5) = 0.25 P(20/20/20) = (0.25)(0.25)(0.25) = 0.0156

  18. P(20 or less 4 times) = (0.75)(0.75)(0.75)(0.75) = 0.316 P(No/No/No/No/WIN) = (11/12)^4 * (1/12) = 0.0588 1 - P(no gold in 6) = 1 – (11/12)^6 = 0.407

  19. Common Errors • Don’t confuse disjoint and independent!! Disjoint events CANNOT be independent!!

  20. Example In a previous WAP problem, we calculated the probabilities of getting various M&Ms. a) If you draw one M&M, are the events of getting a red one and getting an orange one disjoint or independent or neither? b) If you draw two M&Ms one after the other, are the events of getting a red on the first and a red on the second disjoint or independent or neither? c) Can disjoint events ever be independent? Explain.

More Related