Download Randomization / Probability Models

Randomness
What is random?
• Tossing a coin ?
Randomness
What is random?
• Tossing a coin ?
• A phenomenon is called random if individual outcomes are
uncertain, but there is a regular distribution of outcomes
in a large number of repetitions.
•The probability of any outcome of a random phenomenon
is the proportion of times the outcome would occur in a
long series of repetitions.
(Probability describes what happens in many trials)
Randomness
What does this mean ?
• We can never observe a probability exactly.
What is the best way to understand randomness ?
• Observe random behavior
• Long-run regularity
• Short runs. (Unpredictable)
Randomness
How to observe randomness ?
1) You need a long series of independent trials.
2) Computer simulations start with given probabilities.
3) Computer trials are useful for long runs of trials.
The Uses Of Probability
1) Game Theory / Probability Theory
2) Money Raisers
3) Flow of traffic, genetic makeup, rate of return on
risky investments.
Homework
1, 2, 4, 5
Probability Models
What is a probability model ?
1) A list of all possible outcomes.
2) A probability for each outcome.
Sample Space
• A sample space S of a random phenomenon is the set of
all possible outcomes.
Example 1 : Roll a die. What is my sample space ?
S = { 1, 2, 3, 4, 5, 6}
Example 2: Roll two dice and add their totals.
S = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
Events
• An event is an outcome or a set of outcomes of a random
phenomenon. In other words, it is a subset of the sample
space.
Example 3: Let our event be the “sum of two dice being
greater than 8”.
A = { 9, 10, 11, 12}
Example 4 : Let our event be “rolling an even number on
a single die.”
A = { 2, 4, 6}
Events
• An event is an outcome or a set of outcomes of a random
phenomenon. In other words, it is a subset of the sample
space.
Example 5 : Let our event be “rolling a six”.
A={6}
Example : Let our event be “rolling a seven on a single
die”.
A={
} = 
(Empty Set)
Properties O’ Probability Models
1) Any probability is a number between 0 and 1.
Example : Let our sample space be S = { A, B, C, D}
Imagine the probabilities are :
A
0.25
A
- 0.25
B
0.33
B
1.33
C
0.16
C
16
D
0.26
D
0.26
Good
Bad
Properties O’ Probability Models
2) All possible outcomes together must have probability 1.
S = { A, B, C, D}
A
0.25
A
- 0.25
B
0.33
B
1.33
C
0.16
C
16
D
0.26
D
0.26
Properties O’ Probability Models
3) The probability that an event does not occur is 1 minus
the probability that the event does occur.
S = { A, B, C, D}
A
0.25
B
0.33
Q: What is the probability that A
will not occur ?
C
0.16
A: 1 - 0.25 = 0.75
D
0.26
Q: What is the probability that C
will not occur ?
A: 1 - 0.16 = 0.84
Properties O’ Probability Models
4) If two events have no outcomes in common, the
probability that one or the other occurs is the sum of
their individuals properties.
S = { A, B, C, D}
A
0.25
B
0.33
C
0.16
D
0.26
Let E1 be the event of picking A or B.
E1 = { A, B}
Let E2 be the even of picking D
E2 = { D }
Q: What is the probability of picking E1 or E2 ?
A: P(E1) + P(E2) = .58 + .26 = 0.84
Properties O’ Probability Models
1) Any probability in a number between 0 and 1.
2) All possible outcomes together must have probability 1.
3) The probability that an event does not occur is 1 minus
the probability that the event does occur.
4) If two events have no outcomes in common, the
probability that one or the other occurs is the sum of
their individuals properties.
Properties O’ Probability Models
( Fancy, Schmancy Style )
• Rule 1 : The probability P(A) of any event A satisifes
0 < P(A) < 1
• Rule 2 : If S is the sample space in a probability model,
then P(S) = 1
• Rule 3 : The complement of any event A is the event that
A will not occur, written as Ac . The complement
rule states that
P(AC) = 1 - P(A)
• Rule 4 : Two event A and B are disjoint if they have no
outcomes in common, so can never occur
simultaneously. If A and B are disjoint,
P(A or B) = P(A) + P(B)
This is the addition rule for disjoint events.
Venn Diagrams
Draw a picture of your probability model.
1) Start with a rectangle representing your sample space S
2) Let the events represent shapes in your diagram
Notice that these
events are disjoint
Venn Diagrams
What if we have two disjoint events which use the
entire sample space S ?
Venn Diagrams
S = { A, B, C, D, E, F }
E1 = { A, B, C }
E2 = { A, D }
E3 = { E, F }
A
B
C
D
E
F
Assigning Probabilities
Finite Sample Space
1) Assign a probability to each individual outcome.
2) The probability of any event is the sum of the
probabilities of the outcomes of the outcomes in
the event.
Example : S = { A, B, C, D}
A
0.25
B
0.33
C
0.16
D
0.26
E1 = { A, B }
P(E1) = .25 + .33 = .68
E2 = {A, B, D}
P(E2) = .25 + .33 + .26 = .84
P( E1 or E2) = ?
Assigning Probabilities
Equally Likely Outcomes
• If a random phenomenon has k possible outcomes, all equally
likely, then each individual outcome has probability 1/k.
• The probability of any event E is :
P(A) =
=
Count of outcomes in E
Count of outcomes in S
Count of outcomes in E
k
Assigning Probabilities
Equally Likely Outcomes
Example :
S = { A, B, C, D}
A
1/4 = 0.25
B
1/4 = 0.25
C
1/4 = 0.25
D
1/4 = 0.25
Let E1 = {A, C}
P(E1) = 2/4 = .5
Let E2 = {A, B, D}
P(E2) = 3/4 = .75
Independence
• Two events A and B are independent if knowing that one
occurs does not change the probability that the other occurs.
Example : Let’s flip a coin twice. E1 will be the result of the
first toss, and E2 will be the result of the second flip.
Q: Are these independent ?
A: Yes
Example : Let’s draw two cards from a standard deck of
cards. Let E1 be the event of “Drawing the King of Hearts”,
and E2 be the event of “Drawing the Ace of Spades.”
Q: Are these independent ?
A: No
Independence
The Multiplication Rule for Independent Events
• If A and B are independent, then
P(A and B) = P(A) * P(B)
Example: Let E1 be the event of flipping a head on a fair
coin, and let E2 be the event of drawing the 7 of clubs
from a standard deck.
Q: What is the probability of E1 happening and then E2
happening ?
A: Since they are independent,
P(E1 and E2) = P(E1) * P(E2)
= (1/2) * (1/52) = 1/104 = .0096
Notes
1) The multiplication rule holds if A and B are independent,
and not otherwise.
P(A and B) = P(A)*P(B)
2) The addition rule holds if A and B are disjoint, and
not otherwise
P(A or B) = P(A) + P(B)
Homework
10, 13, 14, 15, 16, 17,
20, 21, 23, 26, 27, 32,
34