MATH1P98 - Assignment #2 - Section #4 Due: Monday, October 17 @ 11:59 pm Students are expected to complete all questions on the assignment. However, only a subset of questions will be considered for marking. Marks will be deducted for incomplete assignments. Assignment submissions must be neat, legible, written on one side of the page only, and questions must be submitted in order. A cover page must be attached to the front of the assignment (see sample cover page on Sakai). Staple all pages on the top left corner. Please submit answers to the following questions to your assignment drop box by 11:59pm of the due date. The text is Introduction to Probability & Statistics (3rd Edition), by Mendenhall, et. al. 1. (Law of Large Numbers, p. 134, If you let n, the number of repetitions of the experiment, become larger and larger (n ), you will eventually generate the entire population). Simulate the experiment: rolling of two dice using Excel to find the rolled sum. Follow the instructions: - Open a new Excel document. - Click on cell A1, then click on the function icon fx and select M ath&T rig, then select RAN DBET W EEN . - In the dialog box, enter 1 for bottom and enter 6 for top. - After getting the random number in the first cell, click and hold down the mouse button to drag the lower right corner of this first cell, and pull it down the column until 25 cells are highlighted. When you release the mouse button, all 25 random numbers should be present. - Repeat the steps b) to d) for the second column. - Put the rolled sum of two dice in the third column: Highlight the first two cells in the first row and click on AutoSum icon. Once you receive the sum of two values in the third cell, repeat the step d) to get rolled sum for all 25 values. - Attach the screenshot of your Excel file to your assignment. a) Find the (classical) probability that the rolled sum of two dice is equal to 9. b) Based on the results of experiment (25 trials), estimate the probability (relative frequency approximation) that the rolled sum of two dice is equal to 9. c) Repeat the simulation experiment for 50 and 100 trials, and estimate the probability (relative frequency approximation) that the rolled sum of two dice is equal to 9 for each experiment. Which probability has the closest value to the classical probability? d) Briefly explain how these experiments demonstrate the Law of Large Numbers. In other words, demonstrate that the relative frequency approaches the actual probability for an experiment repeated an infinite number of times. 1 e) Construct the (classical) probability distribution of a rolled sum associated with the rolling of the two dice. 2. (Canadians Are Huge Online Users, problem 4.62, p. 159). According to comScore, a company that provides digital marketing intelligence, Canadians spend more time online when compared to a group of countries that includes the United States, France, and the United Kingdom. In Canada, 68% of the population is online, as compared to 62% in France and the United Kingdom, 60% in Germany, 59% in the United States, 57% in Japan, and 36% in Italy. In fact, users in Canada spent an average of almost 42 hours a month online (or more than 2,500 minutes). Five hundred online users were selected from St. Catharines, Ontario, and were categorized by age and gender as shown in the probability table below. Gender Male (M) Female (F) Total Age Group Under 18 18-35 Years (A) Years (B) 0.21 0.16 0.24 0.18 0.45 0.34 Over 35 Years (C) 0.10 0.11 0.21 Total 0.47 0.53 1.00 If one online user is selected at random from that group of 500 online users, find the following probabilities: a) What is the probability the randomly selected online user is under 18 years of age? b) What is the probability the randomly selected online user is female? c) What is the probability the randomly selected online user is female and under 18 years of age? d) If the randomly selected online user is male, what is the probability he is over 35 years of age? e) If the randomly selected online user is under 18 years of age, what is the probability the online user is female? f) What is the probability the randomly selected online user is male or over 35 years of age? g) If the randomly selected online user is female, what is the probability that she is 18 to 35 years old? 3. The probability that a visit to a primary care physicians (PCP) office results in neither lab work nor referral to a specialist is 35%. Of those coming to a PCPs office, 30% are referred to specialists (event A) and 40% require lab work (event B). a) Draw a Venn diagram and identify the following probabilities: P (A), P (Ac ), P (B), P (B c ), P (A B)c , P (A B). b) Determine the probability that a visit to a PCPs office results in both lab work and referral to a specialist. c) Determine the probability that a visit to a PCPs office will result in lab work given that he/she has referred to a specialist. 2 4. A group of 22 students decided to play soccer, and they need to be divided into two teams of 11. Two people can be goalies, 4 people can play as forwards and the rest can play either defense or midfield. a) In how many ways can two forwards be chosen on each team if there are four forwards available? a) In how many ways can 22 students be divided into two equal groups if each team has one goalkeeper and two forwards? 5. There are 20 apples in a basket: 5 Macintosh and 10 Gala and 5 Spartan. You randomly select two apples. a) Develop a tree diagram of all possible outcomes of this experiment. b) Find the probability that the first apple is Gala and the second is Macintosh. c) Find the probability that at least one apple is Spartan. 6. A family has 3 children. a) Construct the probability distribution for x, the number of girls. b) Find the mean and standard deviation for the number of girls. c) Is it a binomial distribution? Why or why not? 7. According to Statistics Canada (Share of Non-Alcoholic Beverage Market, 2008), 16% of Canadians consume coffee, 12% - tea, 72% - others. Twenty five people have registered for a workshop on potential health hazards associated with consumption of caffeine in food and dietary supplements. a) Find the probability that exactly five people from the list of participants prefer drinking either tea or coffee. Calculate the probability manually and check your answer using Excel. b) Find the probability that at least two people out of 25 prefer drinking coffee. Calculate your answer manually and check your answer using Excel. c) What is the average number of people who prefer tea or coffee? 3