Hints on doing Probability problems: Feb 2, 2017 1. Define your events: A, B, C, etc. (i.e., A = picking a number out of a bag, B = it will rain today, C = a person has a disease) 2. Assign a probability to each of the events using (1) physical properties (number of faces on a die), (2) past performance (weather, last year's batting average), or (3) a guess. (i.e. P(A) = 1/5, P(B) = 0.10, P(C) = 0.02) 3. Determine which mathematical method(s) should be used to obtain the desired final probability a. Addition rule: P(A B) = P(A) + P(B) - P(A B) b. Conditional probability: P(A|B) and P(B|A) P(A|B) * P(B) = P(A B) c. If P(A B) = P(A)*P(B), then A and be are \"independent\" d. If P(A B) = 0, then A and B are \"mutually exclusive\" e. After a marble (card, person, etc) is picked out of a bag, is it put back in the bag before the next pick (with replacement) or is it set aside and no longer used (without replacement)? f. Binomial theorem: 1 = (p + q)n where each of the n events is independent of each other g. Tree diagrams (i.e., for false positive problems) h. Venn diagrams, when given P(A B) and/or P(A B) n! k ! ( nk ) ! i. Combination formula: nCk = j. k. l. m. 6x6 table for 2 dice problems Contingency table Complement rule: P(~A) = 1 - P(A) For a distribution with P(x) (a probability distribution function or PDF) = x*P(x), 2 = (x-)2 * P(x) STAT 200 Section 4035 Spring 2017 Quiz #2a Please answer all questions. The score for each question is posted at the beginning of the question, and the maximum score for Quiz #2a will be 75 points. This score will be added to the 25 points of Quiz #2b. The 100 points will be scaled to 80 quiz points. Make sure your answers are as complete as possible and show your work/argument. In particular, when there are calculations involved, you should show how you come up with your answers. The quiz is due by 11:59 pm, Sunday, February 12, 2017 Eastern Standard Time. IMPORTANT: Per the direction of the Dean's Office, you are requested to include a brief note at the beginning of your submitted quiz, confirming that your work is your own. By typing my signature below, I pledge that this is my own work done in accordance with the UMUC Policy 150.25 - Academic Dishonesty and Plagiarism (http://www.umuc.edu/policies/academicpolicies/aa15025.cfm) on academic dishonesty and plagiarism. I have not received or given any unauthorized assistance on this assignment/examination. _________________________ Electronic Signature 1. (5 points) Use the binomial calculator to do this problem. http://onlinestatbook.com/2/calculators/binomial_dist.html Hopefully by April 2017, I will have 14 descendants - 13 boys and 1 girl. The binomial equation is 1 = (p + q)n. Let p be the probability of having a boy and q be the probability of having a girl. a. What is the value of p? (p = in the binomial calculator.) b. What is the value of n that should be used? c. What is the probability of having 13 boys and 1 girl by chance? (include a screen shot of your answer) d. What is the probability of being \"normal\" (for people with 14 descendants) by having exactly 7 boys and 7 girls? (include a screen shot of your answer) 2. (5 points) Last year McDonalds ran a promotion where it gave out Monopoly stickers with each purchase. If you got a winning sticker, you would get a small extra item free. The average person makes 3 purchases per visit, i.e. a Big Mac, French fries, and a drink; or an egg McMuffin, hash browns, and coffee; etc. Use the binomial calculator to do this problem. a. How many purchases does the average person make in 2 visits? b. What is the value of n that should be used? c. Let the probability of a single sticker being a winner be 10%. What is ? d. What is the probability of winning at least 1 prize (i.e. 1 or more prizes up to 6) after purchasing 6 items? (include a screen shot of your answer) e. What is the probability of winning exactly 1 prize after purchasing 6 items? (include a screen shot of your answer) f. What is the probability of winning 0 prizes after purchasing 6 items? (include a screen shot of your answer) 3. (5 points) Calculate the odds of winning the Power Ball 2nd prize of $1 million. In order to win, you must match the numbers of 5 white balls numbered 1 to 69, and NOT match the red Power ball numbered 1 to 26. Write your final answer in 1/p instead of the fraction p. For example, = 4. 0.10 = 10. a. What is the probability of picking 1 of the 5 winning white balls from the bag? b. What is the probability of picking 1 of the remaining 4 winning white balls from the bag? c. What is the probability of picking 1 of the remaining 3 winning white balls from the bag? d. What is the probability of picking 1 of the remaining 2 winning white balls from the bag? e. What is the probability of picking the last remaining winning white ball from the bag? f. What is the probability of NOT picking the winning red Power ball from another bag? (Note: The white balls and the red Power balls are in different bags. They are not in the same bag.) g. Using the multiplication rule, what is the probability of winning the Power ball 2nd prize in 1/p? (Note: The Power ball ticket says the odds of winning are 1 out of 11,688,053.52. If you do not get something close to this number, then you did something wrong!) 4. (10 points) The probability that a person has a certain fatal disease is 2% of the population. There is a test for detecting this disease. If you have the disease, the test gives the correct result 97% of the time. If you do not have the disease, then the test gives the correct result 95% of the time. Fill in the following tree diagram. a. What is the probability that a person who gets positive test results actually has the disease? (true positive) b. What is the probability that a person who gets positive test results does not have the disease? (false positive) c. Of all of the people who test positive for the disease, what is the probability that they actually have the disease? P(true positive| positive test results) d. You are the medical advisor to the president. Based on the above statistics, would you recommend that everyone in the country get tested for this fatal disease? 5. (10 points) Determine the poker odds of drawing the following hands from a standard 52 card deck. What are the odds of drawing a royal flush? (Drawing AKQJ10 of the same suit.) a. What is the probability of drawing the A of spades for your first card? b. What is the probability of drawing the K of spades for your 2nd card? c. What is the probability of drawing the Q of spades for your 3rd card? d. What is the probability of drawing the J of spades for your 4th card? e. What is the probability of drawing the 10 of spades for your 5th card? f. Now there are 4 suits (spades, hearts, diamonds, and clubs). What number do you need to multiply the above probability to take into account that there are 4 suits? g. Now we drew the cards in a specific order (AKQJ10). But the order of the drawing of the cards does not matter in the end. What number do you need to multiply the above to get the final probability which takes into account that drawing order does not matter? (How many different ways can you arrange 5 different distinct objects?) e.g. KQJ10A, QJ10AK, etc. h. Multiply the above numbers to get the final probability. Express you answer as 1/p (i.e., .10 = 10, .5 = 2) (Note: The odds for getting a flush are 1/p = 649,740. If you did not get this number, then you did something wrong.) 6. (10 points) Use the following values to fill in the Venn diagram. A = \"2\" + \"3\