6'1 Simulation

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6.1 Simulation

• Probability is the branch of math that describes
  the pattern of chance outcomes
• Probability 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

• State the problem or describe the random
  phenomenon.
• State the assumptions.
• Assign digits to represent outcomes.
• Simulate many repetitions.
• 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?

• State the problem or describe the random
  phenomenon (above).
• State the assumptions.
• Assign digits to represent outcomes.
• Simulate many repetitions.
• 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 (exs 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

• You must have a long series of independent
  trials.
• The idea of probability is empirical (need to
  observe real-world examples)
• Computer simulations are useful (to get several
  thousand 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 coinEvent 2 Rolling a
die1) How many outcomes are there? List the
sample space. Tree diagram Rule2) Find the
probability of flipping a head and rolling a 3.
Find the probability of flipping a tail and
rolling a 6.3) of outcomes?
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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
• P(0 heads)
• P(1 head)
• P(2 heads)
• 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
• Find the probability of getting one ace, then 2
  aces without replacement.
• Find the probability of getting one ace, then 2
  aces with replacement.