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# 400Hw03

Course: STAT 400, Spring 2011

School: University of Illinois,...

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STAT 400 Spring 2011 Homework #3 (10 points) (due Friday, February 11, by 3:00 p.m.) No credit will be given without supporting work. 1. During two-and-a-half years of research, bio-psychologist Onur Gntrkn discovered that when people kiss, they turn their heads to the right roughly twice as often as to the left. (Gntrkn, O. Human behaviour: Adult persistence of head-turning asymmetry. Nature, 421, 711,...

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400 Spring STAT 2011 Homework #3 (10 points) (due Friday, February 11, by 3:00 p.m.) No credit will be given without supporting work. 1. During two-and-a-half years of research, bio-psychologist Onur Gntrkn discovered that when people kiss, they turn their heads to the right roughly twice as often as to the left. (Gntrkn, O. Human behaviour: Adult persistence of head-turning asymmetry. Nature, 421, 711, (2003).) Suppose the probability that a person would turn his/her head to the right is 2/3 , and the probability that a person would turn his/her head to the left is 1/3 . A couple is planning a kiss on Valentines day. Assume that their choice of which way to turn their heads is independent of each other. a) b) What is the probability that they would both turn their heads to the right (and kiss)? What is the probability that they would bump noses (i.e., choose the opposite direction to turn their heads)? Similar arguments with 3/4 and 1/4 predict phenotypic ratios of 9:3:3:1 in F 2 offspring in Mendel's dihybrid crosses. 2. From a group of 16 male and 9 female armadillos, Noah must choose two to travel on his ark. Unable to distinguish between male and female armadillos, Noah must choose at random. a) Noah chooses the two armadillos at random. Compute the probability that Noah gets two armadillos of the opposite sex (i.e., one male and one female armadillo). In order to improve his chances of selecting at least one male and one female armadillo, Noah decides to "cheat" and select three armadillos to travel on his ark. Compute the probability that Noah gets at least one male and one female armadillo. b) 3. Does a monkey have a better chance of rearranging ACCLLSUU or AABEGLR to spell ALGEBRA? to spell CALCULUS 4. An electronic device has four independent components. Two of those four are new, and have a reliability of 0.80 each, one is old, with 0.75 reliability, and one is very old, and its reliability is 0.60. a) Suppose that the device works if all four components are functional. What is the probability that the device will work when needed? Suppose that the device works if at least one of the four components is functional. What is the probability that the device will work when needed? Suppose that the four components are connected as shown on the diagram below. Find the reliability of the system. b) c) 0.60 0.80 0.80 0.75 5. Suppose we roll a pair of fair 6-sided dice. Consider the following random variables: X = the sum of the outcomes on the two dice. Y = the maximum largest) (the of the outcomes on the two dice. Construct the probability distribution of each of these random variables and compute their expected values. 6. Sally sells seashells by the seashore. The daily sales X of the seashells have the following probability distribution: x 0 1 2 3 4 a) b) c) d) f(x) 0.25 0.20 0.15 0.10 Find the missing probability f ( 0 ) = P ( X = 0 ). Find the probability that she sells at least three seashells on a given day. Find the expected daily sales of seashells. Find the standard deviation of daily sales of seashells. (continued) Suppose each shell sells for \$5.00. However, Sally must pay \$3.00 daily for the permit to sell shells. Let Y denote Sallys daily profit. Then Y = 5 X 3. 6. e) Find the probability that Sallys daily profit will be at least \$10.00. [ Hint: How many shells must Sally sell to have daily profit of at least \$10.00? ] f) g) Find Sallys expected daily profit. Find the standard deviation of Sallys daily profit. 7. A customer for a home insurance policy owns a \$200,000 home. The probability is 0.1% that the home will be totally destroyed by fire, and the probability is 0.5% that the home will suffer a 50% loss due to fire. If we ignore all other partial losses, what premium should the insurance company charge for a policy just to break even? 8. A box contains 2 green and 3 red marbles. A person draws a marble from the box at random without replacement. If the marble drawn is red, the game stops. If it is green, the person draws again until the red marble is drawn (note that total number of marbles drawn cannot exceed 3 since there are only 2 green marbles in the box at the start). Let the random variable X denote the number of green marbles drawn. a) Find the probability distribution of X. [ Hint: There are only 3 possible outcomes for this experiment. It would be helpful to list these three outcomes ( three possible sequences of colors of marbles drawn ). It may be helpful to make a tree diagram for this experiment. Remember that all ( three ) probabilities must add up to one. ] b) Suppose it costs \$5 to play the game, and the person gets \$10 for each green marble drawn. Is the game fair * ? Explain. [ Hint: Find X = E(X), the average (expected) number of green marbles drawn per game. ] * In gambling or betting, a game or situation in which the expected value of the profit for the player is zero (no net gain nor loss) is commonly called a "fair game." From the textbook: 1.3-20 2.1-14 1.4-6 2.2-6 ( ) 1.5-16 2.3-4

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