Aim: What is the importance of probability?

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Aim What is the importance of probability?
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What is the language of Probability?

• Random is a description of a kind of order that
  emerges in the long run
• We call a phenomenon random if individual
  outcomes are uncertain but there is nonetheless a
  regular distribution of outcomes in a larger
  number of repetitions
• The probability of any outcome of random
  phenomenon is the proportion of times the outcome
  would occur in a very long series of repetitions
• In probability we assume fair even though not
  everything is really fair
• Probability describes what happens in very many
  trials, and we must actually observe many trials
  to pin down a probability.

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What is probability theory?

• Probability theory is the branch of mathematics
  that describes random behavior.
• Mathematical probability is an idealization based
  on imagining what would happen in an indefinitely
  long series of trials.

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Exploring Randomness

• You must have a long series of independent
  trials.
• That is, the outcome of one trial must not
  influence the outcome of any other.
• The idea of probability is empirical
• Simulations start with given probabilities and
  imitate random behavior, but we can estimate a
  real-world probability only by actually observing
  many trials.
• Simulations are very useful because we need long
  runs of trials.
• In situations such as coin tossing, the
  proportion of an outcome often requires several
  hundred trials to settle down to the probability
  of that outcome. The kinds of physical random
  devices suggested in the exercises are too slow
  for this. Short runs give only rough estimates of
  a probability.

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The Uses of Probability

• Probability theory originated in the study of
  games of chance.
• Tossing dice, dealing shuffled cards, and
  spinning a roulette wheel are examples of
  deliberate randomization.
• Probability is used in astronomy, math,
  surveying, economics, genetics, biology etc.
• Although we are interested in probability because
  of its usefulness in statistics, the mathematics
  of chance is important in many fields of study.

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Pop Quiz

1. When is a phenomenon random?
2. What is the probability of an event?
3. What is the probability theorem?
4. Describe three bullet-points of exploring
   randomness?
5. How did the probability theory develop?
6. What are examples of deliberate randomization?

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Answers to Pop Quiz

1. A phenomenon random if individual outcomes are
   uncertain but there is nonetheless a regular
   distribution of outcomes in a larger number of
   repetitions
2. The probability of any outcome is the proportion
   of times the outcome would occur in a very long
   series of repetitions
3. Probability theory is the branch of mathematics
   that describes random behavior.
4. (1) You must have a long series of independent
   trials. (2) The idea of probability is empirical
   (3) Simulations are very useful because we need
   long runs of trials.
5. Probability theory originated in the study of
   games of chance.
6. Tossing dice, dealing shuffled cards, and
   spinning a roulette wheel are examples of
   deliberate randomization.

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Class Work

• Use Table B. We can use the random digits in
  Table B in the back of the text to simulate
  tossing a fair coin. Start at line 109 and read
  the numbers from left to right. If the number is
  0, 1, 2, 3, or 4, you will say that the coin toss
  resulted in a head if the number is a 5, 6, 7,
  8, or 9, the outcome is tails. Use the first 20
  random digits on line 109 to simulate 20 tosses
  of a fair coin. What is the actual proportion of
  heads in your simulated sample? Explain why you
  did not get exactly 10 heads.
• You go to the doctor and she prescribes a
  medicine for an eye infection that you have.
  Suppose that the probability of a serious side
  effect from the medicine is 0.00001. Explain in
  simple terms what this number means.