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MAT 1000 Mathematics in Today's World

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Last Time We discussed the four rules that govern probabilities: 1.Probabilities are numbers between 0 and 1 2.The probability an event does not occur is 1 minus the probability that it does 3.The probability of one or the other of two disjoint events occurring is the sum of their probabilities 4.All possible outcomes together have a probability of 1

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Last Time We also discussed “independent” events. If two events are independent, the probability that both happen is the product of the probabilities of each. We also saw how to analyze simple probabilities using sample spaces, and probability models. Probability models can be represented visually by probability histograms

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Today We will talk about random phenomena with “equally likely” outcomes. Meaning: all outcomes have the same probability. When outcomes are all equally likely, to find probabilities we just need to count.

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Today We will also talk about “simulation.” Probability is a long-term frequency, so we can experimentally approximate probabilities.

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Equally likely outcomes A probability model is a list of all possible outcomes (the sample space) and the probability of each. Models where all outcomes have the same probability are especially easy to work with. When all outcomes in a probability model have the same probability, we say the outcomes are equally likely.

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Equally likely outcomes Some of the simple examples we have considered have equally likely outcomes. Flipping a coin: Rolling a die:

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Equally likely outcomes But, not every probability model has equally likely outcomes. We’ve seen a few examples of this. The gender of a newborn baby: The color of a randomly drawn M&M

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Equally likely outcomes

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“Outcomes” and “events” are somewhat different things An event is a collection of outcomes. When rolling two dice, “the sum of the numbers on the dice is 10” is an event (not an outcome). Which outcomes are in this event?

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Equally likely outcomes

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Probability myths Suppose we flip a coin six times in a row. For each flip, we’ll write down H if we get heads and T if we get tails. Which of these sequences is less likely to happen? H H H T H H T H T Almost everyone would answer that the first sequence is less likely. This is false.

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Probability myths

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Our intuition is wrong in believing that T H H T H T is more likely than H H H But, there is a kernel of truth in this false intuition. It is true that tossing a coin six times and getting heads three times is more likely than tossing a coin six times and getting heads all six times.

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Probability myths Compare the wording: “getting heads three times in six tosses” “tossing a coin six times and getting T H H T H T” These are different statements. Why? There are several ways to toss a coin six times and get heads three times. The sequence T H H T H T is just one way. Here are two more: T T H H T H H H T T T H

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Probability myths

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Simulation This type of question quickly get difficult to analyze. Suppose we want to know the probability of tossing a coin ten times and getting a “run” of at least three heads in a row. This can be calculated mathematically, but it’s fairly subtle. Fortunately, there is an alternative approach. We can estimate probabilities using “simulation.”

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Simulation

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But tossing a coin thousands of times would take a long time. We can use what’s called “simulation” to speed up the process (and this can be done on a computer). The key idea of simulation: use random digits to imitatate chance behavior.

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Simulation

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In our simulation each digit will represent a coin toss. We can let odd digits represent the outcome “heads” and let even digits represent the outcome “tails.” Note this works out because there are 5 odd digits and 5 even digits (including 0), so the probability of choosing an odd number is exactly the probability of getting heads (0.5), and the probability of choosing an even number is exactly the same as the probability of getting tails (0.5 again).

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Simulation Setting up a simulation: Step 3. Simulate many repetitions. We want to simulate what happens when toss a coin ten times. Each digit is one toss. So we look at ten digits from a table of random digits to simulate ten tosses. Let’s start from the first line…

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Chapter 232

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Simulation

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