Probability

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Probability

• Random variable (X)
• Law of large numbers
• Frequency distribution
• Probability distribution

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Random Variables

• Flip a coin. Will it be heads or tails?
• The outcome of a single event is random, or
  unpredictable
• What if we flip a coin 10 times? How many will
  be heads and how many will be tails?
• Over time, patterns emerge from seemingly random
  events. These allow us to make probability
  statements.

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Heads or Tails?

• A coin toss is a random event H or T
  unpredictable on each toss but a stable pattern
  emerges of 5050 after many repetitions.
• The French naturalist, Buffon sic (1707-1788)
  tossed a coin 4040 times resulting in 2048 heads
  for a relative frequency of 2048 /4040 .5069

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Heads or Tails?

• The English mathematician John Kerrich, while
  imprisoned by Germans in WWII, tossed coin 5,000
  times, with result 2534 heads . What is the
  Relative Frequency?
• 2,534 / 5,000 .5068

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Heads or tails?

• A computer simulation of 10,000 coin flips yields
  5040 heads. What is the relative frequency of
  heads?
• 5040 / 10,000 .5040

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• Each of the tests is the result of a sample of
  fair coin tosses.
• Sample outcomes vary.
• Different samples produce different results. But
  the law of large numbers tells us that the
  greater the number of repetitions the closer the
  outcomes come to the true probability, here .5.
• A single event may be unpredictable but the
  relative frequency of these events is lawful over
  an infinite number of trials\repetitions.

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Random Variables

• "X" denotes a random variable. It is the outcome
  of a sample of trials.
• X, some event, is unpredictable in the short
  run but lawful over the long run.
• Randomness is not necessarily unpredictable. Over
  the long run X becomes probabilistically
  predictable.
• We can never observe the "real" probability,
  since the "true" probability is a concept based
  on an infinite number of repetitions/trials. It
  is an "idealized" version of events

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• Playing craps you throw 2 die each throw is a
  random event, each die can come up with a number
  1 thru 6.
• Distribution of 1 die
• Each of 6 numbers has 1/6 probability .167.
• This is a Uniform distribution with probability
  of .167 ? 1.0/6.167

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To figure the odds of some event occurring you
need 2 pieces of information

• Sample space, or a list of all the possibilities
  all the possible outcomes
• The probability of each possible outcome.

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Probability Frequency of Occurrence
Total outcomes
Frequency of occurrence of ways this one
event could happen Total outcomes ways all
the possible events could happen Probability of a
7 is 6 ways out of 36 possibilities
p.167
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Histogram OF OUTCOMES
HTTH HTHT HTTT THTH HHHT TH
TT HHTT HHTH TTHT THHT HTHH TTTT TTTH TTHH
THHH HHHH ----- -- ------- ------- -------
------- X 0 X 1 X 2 X 3 X
4  
Outcome 0 1
2 3
4 Probability .0625 .25
.375 .25 .0625
? S 1.0
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• THE LAW OF LARGE NUMBERS
•         If we observe a large number of outcomes
  of a random variable and then calculate the mean
  of this distribution, this random variable will
  increasingly come close to the true mean of the
  distribution.
• The relative frequency increasingly comes to
  center on the true probability and eventually
  becomes stable.
• ? Over many repetitions the sample mean is
  an unbiased estimator of the population mean, for
  coin tosses ? .5

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• STATISTICAL PROBABILITY
• PERSONAL PROBABILITY

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Much of statistics is based of establishing the
odds, the likelihood, that a single event or
small set of events could have occurred by
chance. EXAMPLE H H H H H -- 5 Heads in 5
tosses -- is possible but rather unlikely. Will
happen on average about ? .55 .03, that is 3
of the time (.5) (.5) (.5) (.5) (.5) .55
.03.
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So, if 100 people toss 5 coins each, 3 out of 100
will get 5 heads or 5 tails in a row. Not so odd
-- well within the realm of possibility in
fact it will routinely happen. To be expected.
So much of what we think is strange, odd,
miraculous is really predictable -- the law of
randomness at work.
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• PERSONAL PROBABILITY
• Aids vs. Heart attack
• Newspapers report the killing of German tourists
  in NYC causing a sharp decline in German tourism,
  
• but the media does not report the number of
  people killed in auto accidents on the way to
  airports.

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The odds of events occurring -- what outcomes
chance alone would produce. How different are
the outcomes you obtain from a given sample
compared to what results you could get solely due
to chance?