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6.2 Probability Models

6.2 Probability Models. All probability models have two components. List of all possible outcomes A probability for each outcome. Sample Space:. · denoted S · the set of all possible outcomes of a random phenomenon (What outcomes CAN occur)

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6.2 Probability Models

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  1. 6.2 Probability Models

  2. All probability models have two components List of all possible outcomes A probability for each outcome

  3. Sample Space: ·denoted S ·the set of all possible outcomes of a random phenomenon (What outcomes CAN occur) ·Example: When flipping a coin, we have a Sample Space of size two. The two possibilities are {Heads, Tails}.

  4. The Good News: We often have some flexibility in defining our sample space, so the choice of S is a matter of convenience as well as correctness. Example: Suppose we’re going to flip a coin four times. One way we could define the sample space is to list all of the possible outcomes - {HHHH, HTHH, etc.}. This would be a sample space of size 16. (Note 2^4 = 16) Another way is that we could just list off the number of heads we get in four tosses. This would be a sample space of size 5, since the outcomes are {0, 1, 2, 3, 4}. We pick our sample space depending on what it is we intend to measure.

  5. To find the sample space 1. Tree diagram 2. List of all possibilities

  6. Tree diagram Example: Menu

  7. List Example: Toss 3 coins

  8. Event: Outcome or set of outcomes of a random phenomenon. EVENT = SUBSET of the Sample Space. Example: Suppose we flip a coin four times, let’s define the Event A to be getting one head out of the four tosses. In which case A = {HTTT, THTT, TTHT, TTTH}.

  9. BASIC RULES OF PROBABILITY 1. ANY probability is a number between 0 and 1. - An event with probability 1 must ALWAYS happen. - An event with probability 0 will NEVER happen. 2. The sum of the probabilities must ALWAYS be 1.

  10. BASIC RULES OF PROBABILITY 3. The probability that an event does not occur = 1 minus the probability that the event does occur. if an event occurs 70% of the time, we know that it DOES NOT happen 30% of the time (because 1 - .7 = .3)

  11. Complement The probability that an event does not occur is called the complement of the event. If A = the event, then Ac = the complement of the event, meaning that A does NOT occur. Also written as A'

  12. Example: Testing Corvettes The General Motors Corporation wants to conduct a test of a new model of Corvette. A pool of 50 drivers has been recruited, 20 of whom are men. When the first person is selected from this pool, what is the probability of not getting a male driver? Because 20 of the 50 subjects are men, it follows that 30 of the 50 subjects are women so, P(not selecting a man) = P(not a man) = P(woman) = 30/50 = 0.6

  13. BASIC RULES OF PROBABILITY 4. If two events have NO outcomes in common, the probability that one or the other occurs is the sum of their individual probabilities.

  14. Suppose one event occurs 40% of the time and another event occurs 25% of the time. As long the two events have NO outcomes in common, together the events happen 65% of the time.

  15. disjoint Two events that have no outcomes in common Also called mutually exclusive

  16. Mathematical Notation for Probability Rules 1. If A = some event, then 0 ≤ P(A) ≤ 1 2. If S = sample space, then P(S) = 1 3. P(Ac) = 1 - P(A) 4. If A, B are disjoint: P(A or B) = P(A)+P(B)

  17. Equally Likely Outcomes: Each outcome has an identical probability Example: Flipping a coin is an equally likely outcome; heads or tails both have probability 0.5.

  18. If a random phenomenon has k possible equally likely outcomes, then each individual outcome has a probability of 1/k. Thus: P(A) = # of outcomes in A = Example: In a table of ten random digits, all outcomes are equally likely so each probability = 1/10. Suppose I wanted to find P(A) where A = {1, 6, 8}. In which case, there are three outcomes in A so P(A) = 3/10 = .3.

  19. Example A sociologist studies social mobility in England by recording the social class of a large sample of fathers and their sons. Social classes are ordered from 1 (lowest) to 5 (highest). Below are the probabilities that the son of a Class 1 father will end up in each of the possible social classes. Son’s Class 1 2 3 4 5 Prob 0.48 0.38 0.08 0.05 0.01 What is the sample space? Are the possible outcomes equally likely? Let A = {son remains in class 1} and B = {son reaches one of the two highest classes} Find: P(A); P(B); P(Ac); P(A or B) P(A or B) P(Ac) P(A) P(B)

  20. 6.2 Part 2

  21. we talked about “disjoint” events - events where there was no overlap in the outcomes. Rule 4 told us: (rule 4) P(A or B) = P(A) + P(B). This answers the question, “What’s the probability that A or B happens?”

  22. Venn Diagram *** Disjoint means that

  23. Venn Diagram worksheet

  24. Now we want to answer the question, “What’s the probability that A and B happen?”

  25. Independence Example: What’s the probability that when we flip a coin twice, we get two consecutive heads? Solution: We’re really trying to find the P(head and head). Experience tells us that the probability of getting a head on the first toss is 0.5 and the probability of getting a head on the second toss is 0.5. So together it’s 0.5 • 0.5 = 0.25 or 1/4. The key point to remember is that getting a head on the first toss does not alter our odds of getting a head on the second toss. This is independence.

  26. Independent Events: The occurrence of one event does not alter the probability of the occurrence of another event. Example: Flipping a coin = indep. events Example: Drawing a card from a deck = dependent events.

  27. Multiplication Rule for Independentevents Rule 5 (multiplication rule) P(A and B) = P(A) • P(B) Note: If you have three independent events, the formula changes logically: P(A and B and C) = P(A) • P(B) • P(C)

  28. Your turn to dazzle me: Does rolling a six-sided die yield independent or dependent events? Independent 2. Find the P(6) on one toss. 1 6 3. Find the P(3 and 4) 0 4. Find the P(37) on one toss. 5. Find the probability of rolling seven consecutive “2s.”

  29. “At Least” A diagnostic test for the presence of AIDS has a probability of 0.005 of producing a false positive - saying you have AIDS when you don’t. Suppose 140 hospital employees are tested and all 140 are free of AIDS. What is the probability that at least one false positive will occur? (Assume independence) Solution: Key phrase = at least P(at least one false positive) = 1 - P(no false positives) Complement Rule = 1 - P(140 negatives) = 1 - 0.995^140 Multiplication Rule = 1-0.496 = 0.504. Conclusion: The probability is greater than 1/2 that at least one of the 140 employees will test positive for AIDS, even though no one has the virus!

  30. Example problems On the TV show, “Who wants to be a millionaire?” contestants answer 15 questions that each have four possible responses. Assume that each question is an independent event. Also, assume that the contestant is totally clueless as to what the answers are. 1. What’s the probability of getting the first question right without any help? 2. What’s the probability of getting all fifteen questions right without any help? Suppose that they’re introducing a new version of the game show whereby the goal is to get the answer wrong. Again, assume each question is an independent event and that the contestant is still clueless. 3. What’s the probability of getting the first question wrong? 4 What’s the probability of getting all fifteen questions wrong? 5. Let A be the event {getting the question right}. What is Ac? 6. What’s the sample space for each game show question? 7. What’s P(S), if S = sample space?

  31. Homework ·6.27-6.33

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