Practice Probability Questions 1. Suppose that you have torn a tendon and are facing surgery to repair it. The orthopedic surgeon explains the risks to you. Infection occurs in 3% of such operations, the repair fails in 14%, and both infection and failure occur together in 1%. What percent of these operations succeed and are free from infection? 2. Consolidated Builders has bid on two large construction projects. The company president believes that the probability of winning the first contract (event A) is 0.6, that the probability of winning the second (event B) is 0.5, and that the joint probability of winning both jobs (event {A and B}) is 0.3. a. What is the probability of the event {A or B} that Consolidated will win at least one of the jobs? b. Are events A and B independent? Do a calculation that proves your answer. c. Draw a Venn diagram that illustrates the relation between events A and B. Write each of the following events in terms of A, B, Ac, and Bc. Indicate the events on your diagram and calculate the probability of each. Consolidated wins both jobs, Consolidated wins the first job but not the second. Consolidated does not win the first job but does win the second. Consolidated does not win either job. 3. A die is loaded so that the number 3 comes up twice as often as any other number. Make a table of the probability distribution for rolling the die once. What should the probabilities add up to? a.What is the probability of rolling a 3? b.What is the probability of rolling an even number? 4. If you draw an M&M candy at random from a bag of the candies, the candy you draw will have one of six colors. The probability of drawing each color depends on the proportion of each color among all candies made. a. The table below gives the probability of each color for a randomly chosen plain M&M: Color Brown Red Yellow Green Orange Probability 0.3 0.2 0.2 0.1 0.1 b. What must be the probability of drawing a blue candy? Blue ? c. What is the probability that a plain M&M is any of red, yellow, or orange? d. What is the probability that a plain M&M is not brown? e. What is the probability that a plain M&M is both yellow and green? 5. Many fire stations handle emergency calls for medical assistance as well as those requesting fire fighting equipment. A particular station says that the probability that an incoming call is for medical assistance is 0.85. a. What is the probability that a call is not for medical assistance? b. Assuming that successive calls are independent of one another, calculate the probability that two successive calls will both be for medical assistance. c. Still assuming independence, calculate the probability that for two successive calls, the first is for medical assistance and the second is not for medical assistance. d. Still assuming independence, calculate the probability that exactly one of the next two calls will be for medical assistance. (Hint: Draw a tree diagram) 6. In the alpha quadrant, the probability of being in the Federation is .35. Of those in the Federation, 20% are Vulcan. What is the probability that a randomly chosen person in the alpha quadrant is a Vulcan and is in the Federation? 7. The suit of 13 hearts (A, 2 to 10, J, Q, K) from a standard deck of cards is placed in a hat. The cards are thoroughly mixed and a student reaches into the hat and selects two cards without replacement. a. What is the probability that the first card selected is the jack? b. Given that the first card selected is the jack, what is the probability that the second card is the five? c. What is the probability of selecting the jack on the first draw and then the 5? d. What is the probability that both cards selected are greater than 5 (when the ace is considered “low”)? 8. A Gallup Poll conducted in November 2002 examined how people perceive the risks associated with smoking. The following table summarizes data on smoking status and perceived risk of smoking. Assume that it is reasonable to consider these data as representative of the U.S. adult population. a. What is the probability that a randomly selected U.S. adult is a former smoker? b. What is the probability that a randomly selected U.S. adult views smoking as very harmful? d. What is the conditional probability that you choose a former smoker, given that the person chosen views smoking as very harmful? e. Are the events “choose a former smoker” and “choose very harmful” independent? How do you know? f. What is the probability that a randomly selected U.S. adult views smoking as very harmful given that the selected individual is a current smoker? g. What is the probability that a randomly selected U.S. adult views smoking as very harmful given that the selected individual is a former smoker? 9. Of the inhabitants of Wilderland, 40% are Hobbits and 60% are humans. Furthermore, 20% of all Hobbits wear shoes and 90% of all humans wear shoes. a. Make a tree diagram to show the breakdown of residents into Hobbits who either do or do not wear shoes and humans who either do or do not wear shoes. Write the appropriate probabilities along the branches and write the appropriate events and their calculated probabilities to the right of the diagram. b. Suppose 10,000 residents are selected at random. About how many would you expect to be Hobbits who wear shoes? c. About how many of the 10,000 would you expect to be shoeless Hobbits? d. About how many of the 10,000 would you expect to be humans who wear shoes? e. About how many of the 10,000 would you expect to be shoeless humans? f. What percentage of the inhabitants who wear shoes are Hobbits? 10. The probability of winning from a scratch-off lottery ticket is 25%. If you buy 8 tickets and each ticket is independent of the others, what is the probability that you will win at least once? 11) Consider the following experiment. The letters in the word PROBABILITY are printed on square pieces of cardboard (same size squares) with one letter per card. The eleven letter cards are then placed in a hat, and one letter card is randomly chosen (without looking) from the hat. A) List the sample space S of all possible outcomes (2 points). S={ } B) Make a table that shows the set of outcomes (X) and the probability of each outcome (4 points). X P(X) C) Consider the following events: C: the letter chosen is a consonant F: the letter chosen falls in the 1st half of the alphabet (i.e. from A to M) List the outcomes in each of the following events and determine their probabilities (8 points): C={ } P(C) = F={ } P(F) = C or F = { } P(C or F) = FC = { } P(FC) = D) Determine if events C and F are independent. Justify your answer and write your response in a complete sentence (4 points). Things to Know: General Addition Rule:___________________ General Multiplication Rule:______________ Addition Rule for Disjoint: ________________ Mult. Rule for Independent: ______________ Two Ways to Test Independence:___________ Def. of Disjoint and in Symbols: __________