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Name: ______________________________ Due Date: HL Math Problem Set 8 (21 points) Probability Distributions, Binomial and Poisson Distributions ________________________ Directions: Show all work in the space provided. Clearly indicate your final answer. You are to work independently on this problem set. You may use a TI-83/84 calculator only. Each question is worth 2 points each or 1 point per part. **Recall: You have a formula packet (under Topic 6) that may will help! Also: Remember that when solving for the expected value (mean) or variance of a discrete random variable there are two very convenient formulas (they are in your packet too, but I will put them here for quick reference!) 𝜎 2 = 𝑉𝑎𝑟(𝑥) = ∑(𝑥𝑖 − 𝜇)2 𝑝𝑖 = ∑ 𝑥𝑖2 𝑝𝑖 − 𝜇 2 𝜇 = 𝐸(𝑥) = ∑ 𝑥𝑖 𝑝𝑖 1. The number of times a certain factory machine breaks down each working week has been recorded over a long period. From this data, the following probability distribution for the number, X, of weekly breakdowns was produced. x 𝑃(𝑋 = 𝑥) 0 0.04 1 0.24 2 0.28 3 0.16 4 0.16 5 0.08 6 0.04 a. Find the mean of X. b. What would be the expected total number of breakdowns that will occur over the next 48 working weeks? 2. An absent minded mathematician is attempting to log onto a computer, which is done by typing the correct password. Unfortunately he can’t remember his password. If he types the wrong password he tries again. The computer allows a maximum of four attempts altogether. For each attempt the probability of success is 0.4, independently of all other attempts. a. Calculate the probability that he logs on successfully. b. The total number of attempts he makes, successful or not, is denoted by X (so that the possible values of X are 1, 2, 3 or 4). Tabulate the probability distribution of X. 3. The probability of a novice archer hitting a target with any shot is 0.3. Given that the archer shoots six arrows, find the probability that the target is hit at least twice. 4. Joseph and four friends each have an independent probability 0.45 of winning a prize. Find the probability that a. exactly two of the five friends win a prize. b. Joseph and only one friend one friend win a prize. 5. A man rolls ten fair dice. He tells you that one of them is a six. What is the probability that there are at least two sixes? 6. Between the hours of 0800 and 2200, cars arrive at a certain petrol station at an average rate of 0.8 cars per minute. Assuming that arrival times are random, calculate the probability that at least 2 cars will arrive during a particular minute between 0800 and 2200. 7. The number of night calls to a fire station in a small town can be modeled by a Poisson distribution with mean 4.2 per night. Find the probability that on a particular night there will be 3 or more calls to the fire station. 8. A shop sells two brands of television sets. The average number of Kubla sets sold per week is 1.8. Sales take place independently at random times. a. Find the probability that exactly 2 Kubla sets are sold in a given week. b. Find the probability that exactly 4 Kubla sets are sold in a given two week period. 9. Analysis of the scores in football matches in a local league suggests that the total number of goals scored in a randomly chosen match may be modeled by the Poisson distribution with parameter 2.7. The number of goals scored in different matches are independent of one another. a. Find the probability that a match will end with no goals scored. b. Find the probability that 4 or more goals will be scored in a match. One Saturday afternoon, 11 matches are played in the league. c. State the expected number of matched with no goals scored. d. Find the probability that there are goals scored in all 11 matches. e. State the distribution for the total number of goals scored in the 11 matches. Find the probability that more than 30 goals are scored in total.