Probability

							Probability
                Our Goals
 Develop a representation for illustrating the
  probabilities and outcomes of a particular
  scenario
 Practice this representation

 Investigate a complex scenario to illustrate
  the power of probability
                The Setting
You are…
 Blindfolded,

 Led into a room,

 Instructed to throw a dart at the far wall
  until you are told to stop,
 Compute the probability of the possible
  outcomes if…
You stop when you hit a balloon
You stop when you hit a balloon
You stop when you hit a balloon
You stop when you hit a balloon
You stop when you hit a balloon
You stop when you hit a square
    You win      You lose
    nothing      $10


    You win $8
You stop when you hit a square

 Win $10   Lose $3


 Win $5
You stop when you hit a square
   +$25   -$15          +$40

   -$10          -$20   +$10

   +$5           -$15
 Computing the Probability
           Heads on   Tails on
           Flip 1     Flip 1

Heads on
Flip 2


Tails on
Flip 2
             Computing the Probability
                               Roll on First Die
Pr(Sum>9)              1   2        3     4        5   6
Pr(Prod is even)
                   1

                   2
   Roll on
   Second          3
   Die
                   4

                   5

                   6
         2 Draws with Replacement
3 Red Balls            First Draw

4 Green Balls

1 Blue Balls


      Second
      Draw
      2 Draws without Replacement
3 Red Balls          First Draw

4 Green Balls

1 Blue Balls


      Second
      Draw
              Area Model
 Notice we can use area to see when we
  multiply or add probabilities
 We can even deal with dependent events

 As we get more familiar, we don’t need to
  be too accurate with scale- so long as we
  keep track of what the probabilities are
    An Application of Probabilities
 Your demographic has a low occurrence of
  AIDS, about 1 in 100,000.
 You take a test for AIDS that is 99.9%
  accurate, and it comes back positive
 Should you be worried? Why or why not?

 What if a second (independent) test came
  back positive?
Scratch Work
                Generalizing
   We’ve now seen how the area model can be
    used to help compute the probabilities of a
    sample space (especially if the space is
    comprised of two distinct events)
                          Heads    Tails
                          on       on
                          Flip 1   Flip 1
              Heads on
              Flip 2

               Tails on
               Flip 2
              Generalizing
 So how might we try to adjust the area
  model so that it can succinctly represent
  more than two distinct events?
 Example: Use the area model to help you
  compute the probability of flipping atleast 2
  heads in 3 coin flips
                              Generalizing
Example: Use the area model to help you
 compute the probability of flipping atleast 2
 heads in 3 coin flips
            Heads    Tails
            on       on
            Flip 1   Flip 1              HT
                                    HH        TT
Heads on                        
Flip 2

 Tails on
 Flip 2
              One way to generalize
                        The first two flips

                   HH         TH                 TT
          H
3rd
Flip      T

       Pr(>1 H)=                   Pr(exactly 1 H)=
       Pr(<3 T)=                   Pr(3H)=
 Computing only one probability
 If you are interested in only one particular
  outcome, it is typically easier to compute
  only that probability and not model the
  entire scenario.
 Let us consider some examples…
    What is the probability that…
   Exactly 3 heads occur after 4 consecutive
    coin flips.
    – How many possible outcomes are there?
    – How many possible ways are there for there to
      be exactly 3 heads?

    Note: Counting can be reinterpreted as the
     number of choices one has (Recall the
     Cartesian product model of multiplication in
     302A)
    What is the probability that…
   Exactly 2 heads occur after 5 consecutive
    coin flips.
    – How many possible outcomes are there?
    – How many possible ways are there for there to
      be exactly 2 heads?
            Birthday Problem
   Task: Compute the probability that at least
    two people in this room have the same
    birthday, Pr(shared).
           Birthday Problem
 Hint: Consider an simpler probability to
  compute that is related, namely: what is the
  probability that no one has a shared
  birthday. Pr(none shared)
 Question: If we can compute this
  probability, how do we find the original
  probability we were asked about?
                     Summary
   Probability of an outcome in a scenario is the
    number of times the particular outcome can occur
    divided by the total number of all outcomes.
   Expected value represents the expected average
    value of the scenario as it is repeated many times.
   We can use an area model to represent the
    probabilities of a scenario
   Sometimes the probability that an outcome does
    not occur is easier to compute then the probability
    that it does.
                  Homework 5
   Exploration 7.18 in the Red Book
    – Parts I, II, V