Math 141 Practice Exam 3 1. An 8-space code is to be made from the 26 letters of the alphabet with repeats allowed. Find the probability that:

Transcription

1 Math 141 Practice Exam 3 1. An 8-space code is to be made from the 26 letters of the alphabet with repeats allowed. Find the probability that: a) the code contains exactly two a's and no other repeats. b) at least two spaces in the code have the same letter. 2. Make a tree diagram for the following information: Some car owners of Hondas and Toyotas were asked where they went for car service. Out of the 120 car owners surveyed, 70 owned a Honda and 50 owned a Toyota. Of the Honda owners, 20% always went to the dealership for car service, 30% sometimes went to the dealership and the others never went there for car service. Of the Toyota owners, 15% always went to the dealership, 25% sometimes went to the dealership and the rest never went there.

2 3. People who bought either a brand A, B or C personal laptop computer were asked if they were happy with their computer. A, B, and C are the events that the person purchased brand A, B or C respectively. H is the event that the person is happy with his/her computer. 40% bought brand A, 40% bought brand B and the rest bought brand C. 70% of brand A buyers were happy, 80% of brand B buyers were happy and 50% of brand C buyers were happy. a) Make a tree diagram for this information. Do not round any answers. b) Find the probability that a randomly selected consumer in the survey was happy and bought brand A. c) Are the events A and H independent? Show numerical support for your answer. d) Find the probability that a happy consumer bought brand B. e) Find the probability that a happy consumer did not buy brand C.

3 4. In a certain species of plant, it is estimated that 60% of specimens have gene A, 80% have gene B, and 15% have neither gene. (A plant can have both genes.) A Venn diagram might help. a) What is the probability that a randomly selected plant has both genes? b) Are the events "a plant has gene A" and "a plant has gene B" independent? You must show numerical support for your answer. c) What is the probability that a randomly selected plant that does have gene B, also has gene A?

4 5. A box contains 3 red and 6 blue balls. A person randomly selects three balls without replacement. X is the number of blue balls selected. a) Write the probability distribution of X. Make a table of each possible value of X and the corresponding probability. Do not convert to decimals and do not reduce. Show the counting expression and the fraction. b) Find E(X), the expected value of X, showing all work. c) If each red ball weighs 1 ounce and each blue ball weighs 1.5 ounces, what is the expected value of the total weight of the balls selected?

5 6. A 5-point quiz was given to a class. The table shows the frequency of each score. Score #of students Find each of the following statistics for the set of scores. Round to 2 decimal places if necessary. Mean Median Mode(s) Standard Deviation Variance 7. a) The odds a certain horse will win a race are 3:2. What is the probability this horse will win? b) The chance of rain is 30%. What are the odds in favor of a dry day (no rain)? 8. Which of the following random variables have a binomial distribution? If not binomial, briefly give a reason. If binomial, give the expected value of the random variable described. a) A shipment of electronic games contains 30 games, 2 of which are defective. Eight different games are randomly selected and tested. X is the number of defective games found among the eight. b) A batch of 30 games contains 2 defective, as in part a. Games are selected and tested one at a time without replacement until a defective game is found. Y is the number of non-defective selected before the defective is found. c) Electronic games are continually produced. Five percent of the games are defective. Twenty games are randomly selected and tested. X is the number of defective games among the 20.

6 9. Tree seedlings planted in a certain forest have a 0.75 probability of surviving a certain length of time. One hundred seedlings are planted. Assume independence. Round all answers to 4 decimal places. a) Find the probability that exactly 75 seedlings survive.. b) Find the probability that the number of surviving seedlings is at least 75 and at most A sample space is S={ x, y, z, w}. Some probabilities are P(y,z)=0.3, P(x,y,z)= 0.5 P(y,w)=0.6. Fill in the distribution table: s w x y z p