Warm Up 1) A t-shirt company makes shirts in sizes S, M, L, XL and XXL and in the colors chartreuse, burgundy, fuchsia and mauve. How many different types of t-shirts does the company make? 2) Alfredo is taking AP English Lit, AP Chemistry, AP Stats, AP Economics, AP French and Sculpture this year. Assuming all classes are offered each period, how many possible schedules can he have? 3) Assume there are 33 students in this class. If a group of 3 students will be randomly selected, how many groups of selected students are possible?

Chapter 5 – Probability and Simulation Section 5.1 focuses on simulation. Section 5.2 covers conventional probability Section 5.3 covers conditional probability. When we produce data by random sampling the laws of probability answer the question: “What would happen if we did this many times?” Probability is the basis of inference.

Basic Probability Concepts Probability is a number between 0 and 1 that describes the proportion of times an outcome will occur in a chance process over a large number of repetitions. Chance processes are predictable in the long run but not in the short run. The outcome of previous events does not affect the probability of the next event. If a coin has landed on heads after 10 consecutive flips the probability it will land on heads on the next flip is 0.5.

Simulations Simulations are a method of imitating chance behavior based on a model. The random events are modeled using coin flips, die rolls, random integer tables or random number generators. The probability found by simulation will match the actual probability of an event after many, many repetitions. More repetitions of a simulation will give a more accurate estimate of the actual probability.

Simulation Example A couple plans to have children until they have a girl or until they have 4 children, whichever comes first. What are the chances that they will have at least one girl among their children? Let a flip of a coin represent a birth, heads = girl, tails = boy (since both outcomes are equally likely the coin is an accurate imitation of the situation) Flip the coin until a head appears or 4 times, whichever comes first. This is one repetition. Simulate the family having children for 10 repetitions. Record the number of times they have a girl.

Simulation Example Using Table D Assume there are 34 students in this section of AP Stats – 15 are men and 19 are women. If Dr. Lutze randomly selects 5 students, what is the probability that 3 or more will be men? Assign 01 to 15 to represent a man being chosen and 16 to 34 to represent a woman. Use Table D starting at line 101 to simulate selecting 5 students: 19223 95034 05756 28713 96409 12531 Selected: 19, 22, 34, 05, 13 2 are men and 3 are women

Simulation Example Using Table D Perform 10 repetitions starting at a ”randomly” selected line each time. Using your data, estimate the probability 3 or more men are chosen.

Simulation Activity Imagine we are going to play a “Quiz Game” for extra credit. Each student will be numbered from 1 to 32. Students will be randomly selected to answer a question, if they are correct they earn one point for their team. What is the probability your table would not be selected at all? – Use simulations to estimate. Assumptions: 32 students in the class. There are 8 tables and 4 students at each table. 20 questions are asked. Use either the RandInt function on your calculator or Table D to run your simulation. Run at least 10 repetitions.

Simulation Activity In 2016 the Chicago Cubs finally won their first World Series since 1908. Many people feel it is highly unlikely for a team to not win a championship in over 100 consecutive years. Describe a simulation you could run to determine the probability for any baseball team to fail to win a championship for 100 years in a row. For your simulation assume that each team has the same probability to win in each year and that there are 30 teams (are repeats allowed?). Run at least one repetition of your simulation, preferably several.