Something to Think About

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1 Probability Facts Something to Think About Name Ohio Lottery information: one picks 6 numbers from the set {1,2,3,...49,50}. The state then randomly picks 6 numbers. If you match all 6, you win. The number of six number combinations is 18,009,460. If you filled out 18,009,460 cards to cover all number combinations, and if you could fill out the cards at the rate of one card per second, it would take you 208 days to complete the task. If you buy 50 Lotto tickets a week, you can expect to win the jackpot once every five thousand years. If you purchased two lotto tickets a week for 110,000 years, you would have a 50% chance of winning the jackpot once during that period of time. There are people who make money selling books and software containing ways to increase your chances of winning. Some advertise things like "techniques that will increase your chances of winning by 100%. Of course, a great way to increase your chances of winning by 100% is to buy two tickets instead of one. Odds against winning the lottery in Ohio: 18 million to 1 You are 18 times more likely to die from flesh-eating bacteria than you are to winning the lottery (odds against dying from flesh-eating bacteria are 1 million to 1). You are 3600 times more likely to die of cancer from eating a peanut butter sandwich a day (odds: 5000 to 1) You are 600 times more likely to die from lightning (odds: 30,000 to 1) You are 28 times more likely to be dealt a royal flush during the opening hand of a card game. (odds: 650,000 to 1) You are 28 times more likely to be killed by a terrorist while traveling out of the country. Guessing Numbers Get your pencil ready to write down a number. Ready? Without hesitating, write down a number from 1 to 4. Fill in the boxes below with the number of students in your class who responded 1, 2, 3, and 4. Number of Students Choosing Each Number One Two Three Four

2 1. What is the total number of responses for your class? 2. Which number was written down most frequently? 3. Studies have shown that, from among the numbers 1, 2, 3, 4, the number 3 is chosen most frequently. If we want to estimate the probability that a person not in your class would choose number 3, how would we use the information in the boxes to do so? 4. Now estimate the probability of students writing down the responses 1, 2, and 4. Spinner Problems: (The spinner consists of a circle cut into 8 equal wedges, labeled clockwise: blue, yellow, red, blue, red, blue, yellow, red. Place a pencil point down in the center of the circle, through a paper clip. Spin the paper clip.) Spin the spinner 30 times and record the color on which the spinner lands each time. If the spinner lands on a line, spin again. Red Blue Yellow Total Spins Your data: 30 Class data: Use your data to answer the following questions. 1. Is yellow more or less likely than red? Why? 2. Is blue more or less likely than red? Why? 3. Estimate the probability of the spinner landing on yellow the next time it is spun. 4. Estimate the probability of the spinner landing on blue the next time it is spun. 5. If you spin this spinner 90 more times, about now many times would you expect it to land on blue?

3 6. Combine your numbers with the rest of the class, and write down the total number of reds, blues, and yellows seen by the entire class. Now, answer 1-5 again. 1. Is yellow more or less likely than red? Why? 2. Is blue more or less likely than red? Why? 3. Estimate the probability of the spinner landing on yellow the next time it is spun. 4. Estimate the probability of the spinner landing on blue the next time it is spun. 5. If you spin this spinner 90 more times, about now many times would you expect it to land on blue?

4 Homework Name Give answers as the EXACT fraction or decimal. If you must round off, round to three significant digits. 1. Suppose that a college student with no prior evidence of HIV gives a blood sample that is tested for the presence of HIV antibodies. The test comes back positive, but the test is not perfect and the results are not always correct. When told of the test results, the student asks a straightforward question: Do I have HIV? Test Positive Test Negative Total HIV No HIV 983,030 Total 16,964 a. Complete the missing parts of the table. b. What is the probability of having HIV and being tested positive? (The test correctly identifies the presence of HIV.) c. What is the probability that a positive test actually doesn't contain HIV antibodies? (Incorrect test results.) d. What is the probability of having a negative test and having no antibodies present? (The test correctly identifies the absence of HIV.) e. What is the probability that a negative test actually does contain HIV antibodies? (Incorrect test results.) f. What is the probability that a random person chosen out of this population will have HIV?

5 2. A very successful football coach once explained why he preferred running the ball to passing it: "When you pass, three things can happen [completion, incompletion, or interception] and two of them are bad." Can we infer that there is a 2/3 probability that something bad will happen when a football team passes the ball? 3. The most important skating event in the Netherlands is the Elfstedentocht, a race over 124 miles of canals through 11 Dutch cities. This race is only held if the entire course is covered by ice at least 8 inches thick. During the 96 years from 1900 to 1995, the race was held 14 times. Based on these data, what is your estimate of the probability that the race will be held next year? 4. You are playing draw poker and are dealt four spades and a heart. (Fifty-two cards in a deck, 13 cards of each type.) If you discard the heart and draw a new card, what is the probability that this new card will be a spade, giving you a flush? (Assume that there are no other players, since it can be shown that your chances do not depend on whether there are other players, as long as you do not know what cards they have been dealt.) 5. a. If a person is randomly selected, find the probability that his or her birthday is January 1st, ignoring leap years. b. If a person is randomly selected, find the probability that his or her birthday is in November. Ignore leap years. 6. In a study of blood donors, 225 were classified as group O and 275 had a classification other than group O. What is the approximate probability that a person will have group O blood? 7. Which of the following values cannot be probabilities? 0, , -0.2, 3/2, 2/3, square root of 2, square root of A Gallup survey found that 228 people brushed their teeth once a day, 672 people brushed twice a day, and 240 people brushed three times a day. If one of the respondents is randomly selected, find the probability of getting someone who brushes their teeth three times a day. What is the probability of selecting someone who brushes two or more times a day?

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