AP Review Probability Distributions 1. Determine the expected value, variance, and standard deviation for the following discrete distribution. x 1 2 3 4 p(x) .3 .4 .2 .1 2. Identify the following variables as discrete or continuous. a) The length of time until recovery from an a tonsillectomy. b) The number of violent crimes per month committed in your city. c) The number of commercial aircraft near-misses per month. d) The number of winners each week in the Texas Lottery. e) Your blood pressure. f) Your height. 3. In a study of sleep patterns of adults, it has been found that 23% of the sleep time is spent in the REM (rapid eye movement) stage. a) If a sleeping adult is observed 5 randomly selected times, find the probability that exactly one of the five observations will be made during REM sleep. b) Find the standard deviation for the number of REM stages observed in groups of five observations. 4. In a study of middle-aged adults (40 to 65 years), it is found that 7.8% suffer from hypertension. A follow-up study begins with a random selection of 40 middle-aged adults. a) Find the probability that exactly one-fourth of the sample suffer from hypertension. b) Find the mean number of hypertension cases found in such groups of 40. c) Find the standard deviation for the numbers of hypertension cases in groups of 40. d) What is the probability that less than 4 cases of hypertension will be found in a group of 40. e) What is the probability that at least 3 cases of hypertension will be found in a group of 40. 5. A stationery store has decided to accept a large shipment of ball-point pens if an inspection of 20 randomly selected pens yields no more than two defective pens. a) Find the probability that this shipment is accepted if 5% of the total shipment is defective. b) Find the probability that this shipment is not accepted if 15% of the total shipment is defective. 6. On the leeward side of the island of Oahu in the small village of Nanakuli, about 80% of the residents are of Hawaiian ancestry (The Honolulu Advertiser , December 26, 1993). a) What is the probability that the first person you meet in Nanakuli is of Hawaiian descent.? The second person? The third person? b) What is the probability that you will meet at least 4 people before you meet someone who is of Hawaiian ancestry? 7. On Waikiki it is estimated that about 4% of the residents are of Hawaiian ancestry. Repeat question #6 for Waikiki residents. 8. At Fontaine Lake Camp on Lake Athabasca in northern Canada, history shows that about 30% of the guests catch lake trout over 20 pounds on a 4-day fishing trip (Athabasca Fishing Lodges, Saskatoon, Canada). Let n be a random variable that represents the first trip to Fontaine Lake camp on which a guest catches a lake trout over 20 pounds. a) Write out the formula for the probability distribution of the random variable n. b) Find the probability that a guest catches a 20 pound trout for the first time on trip number 3. On trip number 2. c) Find the probability that it takes more than 3 trips for the guest to catch a 20 pound lake trout. 9. Police find that on patrol unit gets a 3% arrest record when it sets up a checkpoint for drunk drivers. a) Find the probability that of 200 drivers checked, there will be exactly one arrest. More than 3 arrests? b) What is the expected number of arrests? The standard deviation of arrests? c) What is the probability that the third driver is the first one arrested? That at least 10 drivers are checked before an arrest is made?