University of California, Los AngelesDepartment of StatisticsStatistics 13 Instructor: Nicolas ChristouProbability - examplesExample 1:For each of the following, list the sample space and tell whether you think the events are equally likely.a. Roll two dice; record the sume of the numbers.b. A family has 3 children; record the genders in order of birth.c. Toss 4 coins; record the number of tails.d. Toss a coin 10 times.; record the longest run of heads.Example 2:Suppose the probability that a U.S. resident has traveled to Canada is 0.18, to Mexico is 0.09, and to both countries s 0.04. What is the probabilitythat an American chosen at random hasa. traveled to Canada but not to Mex ico?b. traveled to either Canada or Mexico?c. not traveled to either country?Example 3:Employment data at a large company reveal that 72% of the workers are married, that 44% are college graduates, and that half of the grads aremarried. What is the probability that a randomly chosen workera. is neither married nor a college graduate?b. is married but not a college graduate?c. is married or a college graduate?Example 4:Seventy percent of kids who visit a doctor have fever, and 30% of kids with fever have sore throats. What is the probability that a kid who goesto the doctor has a fever and a sore throat?Example 5:You pick three cards at random from a deck. Find the probability of each event described below.a. You get no a ces.b. You get all hear ts.c. The third card is your first black card.d. You have at least one diamond.Example 6:The soccer team’s shirts have arrived in a big box, and people just start grabbing them, looking for the right size. The box contains 4 medium, 10large, and 6 extra-large shirts. You want a medium for you and one for your sister. Find the probability of each event described.a. The first two you grab are the wrong sizes.b. The first medium shirt you find is t he third one you check.c. The first four shirts you pick are all extra-large.d. At least one of the first four shirts you check is a medium.Example 7:Given the table below, are high blood pressure and high cholesterol independent? Explain.Blood PressureHigh OKHigh 0.11 0.21Cholesterol OK 0.16 0.52Example 8:Suppose that 23% of adults smoke cigarettes. It is known that 57% of smokers and 13% of nonsmokers develop a certain lung condition by age 60.a. Explain how these statistics indicate that lung condition and smoking are not independent.b. What is the probability that a randomly selected 60-year-old has this lung condition?Example 9:The 60 students of a statistics class can be classified as below according to their academic status and whether they live on or off campus.On campus Off campus TotalSophomore 15 5 20Junior 10 5 15Senior 15 10 25Total 40 20 60A student from this class is selected at random. Let the events A, J, C, S be as follows: A = {senior}, J = {junior}, C = {on campus},S = {sophomore}. Find the probability that the student:a. lives on campus.b. is a senior.c. does not live on campus.d. lives on campus and is a junior.e. lives on campus or is a junior.f. neither lives on campus nor is a junior.g. does not live on campus given that he is a senior.h. lives on campus given that he is a senior.i. is a senior and sophomore.j. lives on campus or lives off campus.k. lives on campus and is not senior.l. does not live on campus or is not a sophomore.m. Are the events senior and campus independent?Example 10:There are 2 urns. In the first urn there are 5 green balls and 4 blue balls. In the second urn there are 4 white balls, 2 yellow balls, and 2 greenballs. One ball is drawn at random from the first urn and it is placed in the second urn. Then a ball is drawn at random from the second urn.a. Find the probability that the ball drawn from the second urn is green.b. Given that the ball drawn from the second urn is green what is the probability that the ball drawn from urn 1 was also green.Example 11:A machine that produces parts for cars is in good condition 90% of the time. When the machine is in good condition 1% of the parts are defective.When it is not in good condition 15% of the parts are defective.a. Find the probability that a randomly selected part is defective.b. Suppose that a randomly selected part was found to be defective. What is the probability that the machine was not in good condition?Example 12:The marketing manager of a toy manufacturing firm is planning to introduce a new toy into the market. In the past, 40% of the toys introducedby the company have been succesful and 60% have not been succesful. Before the toy is actually marketed, market research is conducted and areport, either favorable or unfavorable, is compiled. In the past, 80% of the succesful toys received favorable reports and 30% of the unsuccesfultoys also received favorable reports.a. What proportion of the new toys receive favorable market research r eports?b. Suppose that market research gives a favorable report on a new toy. What is the probability that the new toy will be succesful?Example 13:An unbiased die in the shape of regular dodecahedron has twelve faces with the numbers 2,2,4,4,4,6,6,10,10,10,12,12, shown separately on the faces.The result of a throw is the number shown on the uppermost face. Each of four players throws the die twice and scores the sum of the two results.a. What is the probability that a player will get a score greater than six? (Ans. 0.8888).b. What is the probability that exactly one of the four players will get a score greater than six? Assume that their scores are independentfrom one another. (Ans. 0.0049).Example 14:The color of a person’s eyes is determined by a single pair of genes. If they are bot h blue-eyed genes, then the person will have blue eyes; if theyare both brown-eyed genes, then the person will have brown eyes; and if one of them is blue-eyed gene and the other a brown-eyed gene, then theperson will have brown eyes (because of the latter fact we say that the brown-eyed gene is dominant over the blue-eyed one). A newborn childindependently receives one eye gene from each of its parents and the gene it receives from a parent is equally likely to be either of the two gnes ofthat parent. Suppose that person A and both of his parents have brown eyes, but person A’s sister has blue eyes.a. What is the probabilty that person A possesses a blue-eyed gene? (Ans.23).Suppose person A’s wife has blue eyes.b. What is the probability that their first child will have blue eyes?
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