Random Phenomena

1
Random Phenomena

• Outcome is unknown before the event
• Flipping a coin
• Rolling a die
• Taking a sample from a population
• Long term behavior is predictable
• 50 heads
• 16.67 for 1, 2, 3, 4, 5, or 6
• Sampling Distribution (Chapter 18)

2
Probability

• Subjective (Personal)
• Based on feeling or opinion.
• Empirical
• Based on experience.
• Theoretical (Formal)
• Based on assumptions.

3
Chips Example

• Bag o chips (poker chips).
• Some are red.
• Some are white.
• Some are blue.
• Draw a chip from the bag.
• We dont know how many of each of the three
  colors are in the bag.
• Assumption Each chip has equal probability of
  being chosen (Same size, same weight, same
  texture, etc.)

4
The Deal

• Possible out comes of the draw
• Draw a blue chip win 3 bonus points.
• Draw a red chip win 2 bonus points.
• Draw a white chip lose 3 bonus points.

5
Is this a good deal?

• Subjective (personal) probability
• Based on your beliefs and opinion.
• Empirical probability
• Based on experience.
• Conduct a series of trials.
• Each trial has an outcome (R, W, B).

6
Empirical Probability

• Look at the long run relative frequency of each
  of the outcomes after choosing n50 with
  replacement.
• Blue
• Red
• White

7
Theoretical Probability

• Look in the bag and see how many
• Blue chips
• Red chips
• White chips
• Assumption
• Each chip has the same probability of being
  chosen. Equally likely.

8
Law of Large Numbers

• For repeated independent trials, the long run
  relative frequency of an outcome gets closer and
  closer to the true probability of the outcome.
• How does this compare with the Law of Averages?

9
Law of Large Numbers

• Probability is a long term number
• Ex. Flip a coin 5 times and get 5 heads in a row,
  is a tail due on next flip?
• Random events do not compensate for short term
  behavior
• Over a long sequence of flips, even after a
  sequence of many heads in a row, P(tails after
  sequence) 0.5

10
Law of Large Numbers

• Over the long term, P(heads) 0.5
• Long term - Infinite

11
Probability Rules

• A probability is a number between 0 and 1.
• Something has to happen rule.
• The probability of the set of all possible
  outcomes of a trial must be 1.

12
Probability Rules

• Event a collection of outcomes.
• Win bonus points (Blue or Red chip)
• Complement rule
• The probability an event occurs is 1 minus the
  probability that it doesnt occur.
• P(A) 1 P(AC)

13
Probability Rules

• Disjoint events no outcomes in common.
• Addition Rule for disjoint events.
• P(A or B) P(A) P(B)
• P(Blue or Red) P(Blue) P(Red)

14
Probability Rules

• Independent outcomes - The outcome of one trial
  does not influence the outcome of the other.
• 1) Coin flips as Head and 2) Coin flips as
  Head.
• INDEPENDENT The outcome of flipping a coin does
  not depend on the previous coin flip outcome.
• 1) It snows or not and 2) Class is cancelled or
  not
• NOT INDEPENDENT The outcome of 2) depends on the
  outcome of 1).

15
Probability Rules

• Independent trials
• Multiplication rule for independent trials.
• P(1st outcome and 2nd outcome)
  P(1stoutcome)P(2nd outcome)

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Example

• What is the chance that two people in a row win
  bonus points?
• P(win 1st and win 2nd)P(win
  1st)P(win 2nd)
• P(win 1st) P(Blue or Red) P(Blue)P(Red)
• P(win 2st) P(Blue or Red) P(Blue)P(Red)

17
Three Terms to Look For.

• Not
• This means the compliment that is subtract form
  1.
• Or
• This means to add the probabilities.
• And
• This means to multiply the probabilities.