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Unit 6 Probability & Simulation: the Study of randomness Simulation Probability Models General Probability Rules Questions of chance Successive coin tosses Having a girl among my four children Cereal Box Sports cards Airline Overbookings Answering questions of chance Estimate the likelihood by actually observing the phenomenon many times. Develop a probability model and calculate a theoretical answer. Simulate multiple repetitions. Simulation On the AP Exam, you may receive full credit for solving a probability problem by simulation even if the problem does not call for it! 2 Types of Simulation 1. Simulation with a Fixed Number of Trials Number of trials determined by the problem. Binomial probability 2. Wait-time Simulation We continue to simulate until a certain condition is met. Fixed Number of Trials Simulation Ex 6.3 Coin Tossed Ten Times 1. State the problem or describe the random phenomenon. 2. State the assumptions. (individual likeliness, independent trials) 3. Assign digits to represent outcomes. Are repeated digits okay? 4. Simulate many repetitions. 5. State your conclusions. Wait-time Simulation Ex. 6.6 Girl or Four 1. State the problem or describe the random phenomenon. State stopping rule. 2. State the assumptions. (individual likeliness, independent trials) 3. Assign digits to represent outcomes. Are repeated digits okay? 4. Simulate many repetitions. 5. State your conclusions. Section 6.2 Probability Models Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run. Probability is used to predict future behavior. The idea of probability Coin tossing Count Buffon in the 1700's 4040 times 2048 heads for 0.5069 John Kerrich imprisoned in WWII 10,000 times 5067 heads for 0.5067 Randomness Probability theory is the branch of mathematics that describes random behavior. 1.Random does not mean haphazard. 2. You must have a long series of independent trials. 3. Probability is empirical, meaning it is based on observation of many trials rather than theorizing. 4.Computer simulations are useful as they provide many trials, very fast. Sample Space Being able to properly enumerate the outcomes in a sample space is critical to determining probabilities. Three techniques to make sure you don’t accidentally overlook any outcomes: 1.Tree diagram 2.Multiplication principle 3.Organized or (Systematic) List Tree Diagram Sometimes it's helpful to use a tree diagram. Tree diagrams are good for relatively small sample spaces. What is the sample space for tossing a coin and rolling a die? Tree Diagram Multiplication Principle or Fundamental Counting Principle What is the sample space when we record four coin tosses in order? Organized List What is the sample space when we roll two dice? Sample Space Sometimes the sample space varies depending on what exactly you're asking for! What is the sample space when we want to know the number of heads in four coin tosses? Discrete vs. Continuous Models Discrete models have a countable number of outcomes. Ex: Flipping a coin Continuous models have an infinite number of outcomes. They correspond to intervals on the number line. Ex: heights of a sample of 15 ninth-graders Ex: all the numbers between 0 and 1 S = {all numbers between 0 and 1} Sampling With and Without Replacement Sampling With Replacement: the same outcome can occur more than once in a sample Ex: Draw 5 numbers from a hat, after each number is drawn, put the number back in the hat. Sampling Without Replacement: the same outcome cannot occur more than once in a sample Ex: Draw 5 numbers from a hat, after each number is drawn, do not put the number back in the hat. Sampling With and Without Replacement How many 3 digit numbers can you make with replacement? How many 3 digit numbers can you make without replacement? Use the context of the problem to determine whether the selection is with or without replacement. Counting Problems How many ways can you roll two dice and get a sum of 3? What are the outcomes? How many ways can you roll two dice and get a sum of 7? What are the outcomes?