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VOCABULARY Random Phenomenom ... we know the possible outcome but not the particular one which occur Trial ... a single realization of a random phenomenon Outcomes ... the value of the trial Event ... a collection of outcomes Sample Space ... the collection of all possible outcome values VOCABULARY (more) Law of Large Numbers (LLN) ... the long-run relative frequency of repeated independent events gets closer to the true relative frequency as the number of trials increases Independent events (slightly different than text definition) ... two events are independent if the occurrence on nonoccurrence of one of the events does not change the probability of the other event occuring or not occuring Probability ... a number between 0 & 1 inclusive which reports the likelihood of an event's occurrence. VOCABULARY (more) Probabilities: empirical ... from a real-life long-run relative frequency theoretical ... from a model (think math) personal ... a gut feeling (not in this course!) PROBABILITY RULES (memorize) ... events denoted by A, B, C, D, E, F, ... and S for the sample space ... P(A) in the text and Pr(A) for the probability of event A ... Pr(E) is always between 0 and 1 ... Pr(AC) = Pr(A') = Pr(~A) is the probability that A does not occur; the AC, A', and ~A denote the complement of A ... Pr(AC) = 1 - Pr(A) and Pr(A) = 1 - Pr(AC) ... Pr(AC) + Pr(A) = 1 ... Pr(S) = 1 ... Pr(A or B) = Pr(A) + Pr(B) if A and B disjoint events or mutually exclusive (no overlap) ... Pr(A and B) = Pr(A)*Pr(B) if A and B are independent Some problems with rolling a single die: 1) Pr(4) = 2) Pr(not 4) = 3) Pr(2 or 7) = 4) Pr(5 and 6) = Some problems when rolling two dice, one red and one green, and finding the sum. 1) Pr(2) = 2) Pr(not 2) = 3) Pr(7 or 9) = 4) Pr(7 and 9) 5) Pr(green die is 4 and red die is 6) = red 1 2 3 4 5 6 green _____________________________________________ 1 | 1 1 1 2 1 3 1 4 1 5 1 6 | 2 | 2 1 2 2 2 3 2 4 2 5 2 6 | 3 | 3 1 3 2 3 3 3 4 3 5 3 6 | 4 | 4 1 4 2 4 3 4 4 4 5 4 6 | 5 | 5 1 5 2 5 3 5 4 5 5 5 6 | 6 | 6 1 6 2 6 3 6 4 6 5 6 6 Exercises from text: #2) List the sample space and state whether events are equally likely. a) roll two dice and record the sum b) a family has 3 children; record each child's sex in order of birth c) toss four coins; record the number of tails d) toss a coin 10 times; rcord the longest run of heads #4) The weather reporter on TV makes predictions such as a 25% chance of rain. What do you think is the meaning of such a phrase? What kind of probability is this? #7) A batter who had failed to get a hit in seven consecutive times at bat then hits a game-winning home run. When talking to reporters afterward, he says he was very confident that last time at bat because he knew he was "due for a hit." Comment on his reasoning. #10) On January 20, 2000, the International Gaming Technology company issued a press release: (LAS VEGAS, Nev.) Cynthia Jay was smiling ear to ear as she walked into the news conference at The Desert Inn Resort in Las Vegas today, and well she should. Last night, the 37-year-old cocktail waitress won the world's largest slot jackpot -- $34,959,458 -- on a Megabucks machine. She said she had played $27 in the machine when the jackpot hit. Nevada Megabucks has produced 49 major winners in its 14-year history. The top jackpot builds from a base amount of $7 million and can be won with a 3-coin ($3) bet. a) How can the Desert Inn afford to give away millions of dollars on a $3 bet? b) Why did the company issue a press release? Wouldn't most businesses want to keep such a huge loss quiet? #16) Although it's hard to be definitive in classifying people as right or left handed, some studies suggest that about 14% of people are left handed. Since 0.14*0.14 = 0.0196 = 1.96%, the Multiplication Rule might suggest that there's about a 2% chance that a brother and a sister are both lefties. What's wrong with that reasoning? #20) In a large Introductory Statistics lecture hall, the professor reports that 55% of the students enrolled have never taken a Calculus course, 32% have taken only one semester of Calculus, and the rest have taken two or more semesters of Calculus. The professor randomly assigns students to groups of three to work on a project for the course. What is the probability that the first groupmate you meet has studied: a) two or more semesters of Calculus? b) some Calculus? c) no more than one semester of Calculus?