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6.1 Simulation Probability is the branch of math that describes the pattern of chance outcomes – It is an idealization based on imagining what would happen in an infinitely long series of trials. Probability calculations are the basis for inference Probability model: We develop this based on actual observations of a random phenomenon we are interested in; use this to simulate (or imitate) a number of repetitions of the procedure in order to calculate probabilities (Example 6.2, p. 393)

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Simulation Steps 1)State the problem or describe the random phenomenon. 2)State the assumptions. 3)Assign digits to represent outcomes. 4)Simulate many repetitions. 5)State your conclusions.

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Ex: Toss a coin 10 times. Whats the likelihood of a run of at least 3 consecutive heads or 3 consecutive tails? 1)State the problem or describe the random phenomenon (above). 2)State the assumptions. 3)Assign digits to represent outcomes. 4)Simulate many repetitions. 5)State your conclusions.

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6.2 Probability Models Chance behavior is unpredictable in the short run but has a regular and predictable pattern in the long run! Random is not the same as haphazard! Its a description of a kind of order that emerges in the long run. The idea of probability is empirical. It is based on observation rather than theorizing = you must observe trials in order to pin down a probability! The relative frequencies of random phenomena seem to settle down to fixed values in the long run. – Ex: Coin tosses; relative frequency of heads is erratic in 2 or 10 tosses, but gets stable after several thousand tosses!

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Example of probability theory (and its uses) Tossing dice, dealing cards, spinning a roulette wheel are all examples of deliberate randomization Describing… The flow of traffic A telephone interchange The genetic makeup of populations Energy states of subatomic particles The spread of epidemics Rate of return on risky investments

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Exploring Randomness 1)You must have a long series of independent trials. 2)The idea of probability is empirical (need to observe real-world examples) 3)Computer simulations are useful (to get several thousands of trials in order to pin down probability)

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Sample space for trails involving flipping a coin = Sample space for rolling a die = Probability model for flipping a coin = Probability model for rolling a die =

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Event 1: Flipping a coin Event 2: Rolling a die 1) How many outcomes are there? List the sample space. Tree diagram: Multiplication Rule 2) Find the probability of flipping a head and rolling a 3: Find the probability of flipping a tail and rolling a 6:

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… 1)If you were going to roll a die, pick a letter of the alphabet, use a single number and flip a coin, how many outcomes could you have? 2) As it relates to the experiment above, define an event and give an example:

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Sample space as an organized list Flip a coin four times. Find the sample space, then calculate the following: 1)P(0 heads) 2)P(1 head) 3)P(2 heads) 4)P(3 heads)

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Sampling with replacement: If you draw from the original sample and put back whatever you draw out Sampling without replacement: If you draw from the original sample and do not put back whatever you drew out! EXAMPLE: 1)Find the probability of getting one ace, then another ace without replacement. 2)Find the probability of getting one ace, then another ace with replacement.

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