# Probability by liwenting

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```									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
Flip 1     Flip 1

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
 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
 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)
on       on
Flip 1   Flip 1
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?
compute the probability of flipping atleast 2
Generalizing
compute the probability of flipping atleast 2
on       on
Flip 1   Flip 1              HT
HH        TT
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

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
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
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

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