ece 450 lecture 2 - csun.edudvanalp/ece 450/ece 450 lectures/ece... · ece 450 d. van alphen 1 ece...

ECE 450 D. van Alphen 1

ECE 450 – Lecture 2

• Recall: Pr(A B) = Pr(A) + Pr(B) – Pr(A B) in general

= Pr(A) + Pr(B) if A and B are m.e.

Lecture Overview

– Conditional Probability, Pr(A | B)

– Total Probability

– Bayes’ Theorem

– Independent Events

– Compound Experiments

– Binomial Distribution


Conditional Probability

• Defn: The conditional probability of event A, given that

event B has occurred is

• Note (by symmetry of the definition):

• From (1) & (2), we have two new ways of writing Pr(A B):

Pr(A B) = Pr(A) Pr(B | A) = Pr(B) Pr(A | B)

0)B(P,)BPr(

)BAPr()B|APr(

0)A(P)APr(

)BAPr()A|BPr(

(1)

(2)


Verification Example

(Is the definition reasonable?)

• Experiment: Toss a single die, and find Pr(2 | even)

• Another way to look at it - Let B be the event of getting an

even number:

3/163

61

)evenPr(

)2Pr(

)evenPr(

)even2Pr()even|2Pr(

SBB’

2 4

6

1

3

5

B is called the restricted

sample space; 2 is now

one of 3 equally likely

outcomes.


Another Example

• Experiment: Toss 2 coins

• Sample Space: S = {HH, HT, TH, TT}

• Find the conditional probability of obtaining two heads when

flipping two coins, given that at least one head was obtained;

i.e., find Pr(2 heads | at least 1 head)

• Def: A: event of obtaining 2 heads = {HH}

B: event of obtaining at least one head = {HH, HT, TH}

• Then

3/143

41

)BPr(

)APr(

)BPr(

)BAPr()B|APr(


Side Note & Definition

• Note: Conditional probabilities are themselves probabilities;

thus, they satisfy all the axioms for probabilities:

1. Pr(A | B) 0

2. Pr(B | B) = 1

3. Pr(A C | B) = Pr(A|B) + Pr(C|B) if A and C are m.e.

• Another definition: consider a collection of subsets, {Ai},

A1 A2S

. . . An

(i = 1, …, n ), of S. The collection is

a partition of S if:

Ai Aj = f, i j

U Ai = S, i = 1, …, n


Total Probability Theorem

• Let {Ai} be a partition of S, and let B be a subset of S:

• Then Pr(B) = Pr[ (A1 B) (A2 B) . . . (An B) ]

= Pr(A1 B) + Pr(A2 B) + … + Pr(An B)

Pr(B) = Pr(B|A1)Pr(A1) + Pr(B|A2)Pr(A2) + … + Pr(B|An)Pr(An)

A1 A2S

. . . An

B

A3 Strategy: To find Pr(B),

break apart B, into

mutually exclusive pieces

m.e(see box,

bottom of p.2)


Example Using “Total Probability”

Transistor types X , Y and Z make up 45%, 25%, and 30% of the total number of transistors in a box, respectively. Let B be the event that a transistor fails before 1000 hrs.

Given the reliability information:

Pr(B|X) = .15

Pr(B|Y) = .4

Pr(B|Z) = .25

Find the probability that a randomly chosen transistor from the box fails before 1000 hours.

Pr(B) = ______ _____ + ________ ____+ _______ _____

= _____ ______ + ______ ____+ ______ ______

= .243


Bayes’ Rule: Note & Example

• Note: Use Bayes’ Rule when asked to find some Pr(A|B),

but it would be easier to find Pr(B|A). (“Backwards

conditional probability”)

Example: Say an observed transistor (from the previous

example) fails before 1000 hrs. Find the probability that it

was a Type Z transistor.

Let B: event that transistor fails before 1000 hrs.

We want: Pr(Z|B), non-trivial

We know Pr(B|Z) = .25 (the easier problem; given on p. 7)

Using Bayes’ Rule, next page:


Bayes’ Rule Example, continued

30864.)BPr(

)ZPr()Z|BPr()B|ZPr(

Answer from p. 7 example

• Two cards are drawn without replacement from a 52-card

deck.

• Find the probability that the 2nd is a queen, given that the 1st

is a queen.

• Find the probability that both the 1st and 2nd are queens.

• Unordered answers: 1/221, 3/51

In-Class Practice


Bayes’ Rule In-Class Example

A medical test for a particular type of cancer has the following

properties:

• It correctly detects the cancer (when present) with probability

95%;

• It incorrectly “detects” the cancer (when there is no cancer

present) with probability 20%.

Suppose that this particular type of cancer is present in only

1% of people of your age/sex/ethnicity.

Find the probability that you actually have this cancer, given

that your test is positive (i.e., cancer was detected).

Let c: event that you have cancer.

Let d: event that cancer is detected.

Pr(d|c) = .95; Pr(___|___) = .05 (complements)

Pr(d|c’) = .2; Pr(d’|c’) = _______;

Pr(c) = .01 (a priori) Find Pr(c|d)

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________

_______________________________________________



(Statistically) Independent Events

• Defn: 2 events A and B are statistically independent (__)

if and only if

Pr(A|B) = Pr(A)

(knowing B has occurred tells me nothing about whether or

not A has occurred)

• Equivalently:

Pr(B|A) = P(B)

and Pr(A B) = Pr(A) Pr(B)

(caution: only true for independent events)


Example Using Independence Definition

• Experiment: Toss 2 dice. Define the events

A: sum = 7

B: 1 face = 6

• Pr(A | B)

• Pr(A) = 6/36 = 1/6

Question: Are A, B,

)}6,6()2,6()1,6Pr{(

)}6,1()1,6Pr{(

)Pr(

Pr

B 11

2

36/11

36/2

A, B not _____

A, B dependent


Facts/Thoughts About Independence

• Fact 1: Complementary events are dependent events.

• Fact 2: Unions, intersections & complements of independent events are independent.

• Consider the 2012 election (Obama/Biden vs. Romney/Ryan):

– Event A: Obama wins as Pres.

– Event B: Romney wins as Pres.

– Event C: Biden wins as VP.

– Event D: Ryan wins as VP.

• Let A: it rains today; B: it rains tomorrow. Are A and B independent ?

Is there any pair of

events from this set

that is an independent

pair?


Example Using Independence

• Say the switches in the circuit below are closed ( -ly of each other) at any time with probability 0.1. Find the probability of a closed path from point A to point B.

• Labeling for the Solution:

A B

A B

top

bottom

rightNote: for a closed path to

exist, the “right” switch

must be closed; and,

either the “bottom” switch

must be closed or both of

the “top” switches must

be closed.


Example, continued

Pr(closed path) = Pr[‘right’ and (‘top’ or ‘bottom’)]

= Pr(right) Pr(top or bottom), by Fact 2, p. 14

= (0.1) [Pr(top) + Pr(bottom) – Pr(both top & bottom)]

by Corollary 4

= .1 [(.01) + .1 – (.01)(.1)] = .0109

(both switches)

(both switches: (.1)(.1) )

Note: Don’t round off!

A B

top

bottom

right


Compound Experiments

• Consider experiments: Ea, Eb

with sample spaces: Sa, Sb

• Say Sa = {a1, …, an} and Sb = {b1, …, bm}.

• Define Sa x Sb (the Cartesian Cross Product) as the set of ordered pairs with the 1st element from Sa and the second element from Sb

• Do experiments Ea & Eb (jointly), and get pairs of outcomes from Sa x Sb

• The joint performance of Ea & Eb is said to be a compound experiment.


Cross Product Examples & Bernoulli Trials

Example 1: Toss a coin 2 times, with S1 = S2 = {H, T}.

S = S1 x S2 = { ____, ____, _____, _____}

Example 2: Toss a coin 3 times, with S1 = S2 = S3 = {H, T}.

S = S1 x S2 x S3 = { (HHH), (HHT), (HTH), (HTT), (THH),

(THT), (TTH), (TTT) }

Definition: A Bernoulli Trial is an experiment with only 2 possible outcomes, sometimes called “success” and “failure”.

Success 1 yes, Failure 0 no


Binomial Experiments

• Definition: A Binomial Experiment is an experiment consisting of n independent Bernoulli Trials.

– Let Ak denote the event of getting k successes (and thus n-k failures) in n trials.

– Let p be the probability of success on each trial.

– Let q = 1 – p be the probability of failure on each trial.

• Example: Consider a 5-trial binomial experiment, and say we are interested in the probability of having 2 successes (first), followed by 3 failures.

– Pr{1, 1, 0, 0, 0} = __ __ __ __ __ = ______

(O.K. to multiply since the events are ____)


Binomial Experiments, continued

• Refined example: Say we want to know the probability of

getting 2 successes, in 5 independent trials (order doesn’t

matter):

Pr{A2} = p2 q3

• In general, the probability of getting k successes in n

(independent) Bernoulli tials is

pn(k) = Pr{Ak} = pk qn – k (Binomial Experiments)

2

5

(since there are 5C2 ways to decide where to

put the 2 successes in the string of 5 outcomes)

k

n


Binomial Experiment Examples

1. 10 missiles are fired at a tank; each missile has a (0.2) probability of hitting the tank, independently of the other missiles. Find the probability that exactly 3 missiles hit the tank.

p10(3) = Pr{ A3} = (__)3 (__)7 .2013

2. A certain football player can catch 2/3 of the passes thrown to him. He needs to catch at least 3 more passes for his team to win the game. Find the probability that his team wins if the quarterback throws to him 5 more times.

Pr{win} = Pr{at least 3 catches}

= Pr{exactly 3 catches or exactly 4 catches orexactly 5 catches}


Binomial Experiment Examples, continued

= Pr{A3 A4 A5}

= Pr{A3} + Pr{A4} + Pr{A5}

= ___ ___ ___ + ___ ___ ___ + ___ ___ ___

= .790123

** Note: we can add these probabilities, because the events

“catch exactly 1 pass”, “catch exactly 2 passes”, and “catch exactly 3 passes” are:

_____________________________ events.


Binomial Experiment Examples

3. Using a normal deck of cards, say we cut the deck 5 times; find the probability of getting an ace on at least 3 cuts.

Pr{at least 3 aces} = Pr{exactly 3 aces or exactly

4 aces or exactly 5 aces)

(Bernoulli Trials)

= Pr{A3} + Pr{A4} + Pr{A5}

=

= .00404

m.e.

051423

13

12

13

1

5

5

13

12

13

1

4

5

13

12

13

1

3

5

513

1501


Danger!!

1. Pr(A B) = Pr(A) + Pr(B) – Pr(A B)

= Pr(A) + Pr(B) if A, B m.e.

2. Pr(A B) = Pr(A) Pr(B | A)

= Pr(A) Pr(B) if A, B independent

Review

• Pr(A|B) = ___________________ (defn)

= ___________________ (Bayes’ Rule)

• Total Probability: If {Ai} is a partition of sample space S, then

Pr(B) = ___________________________________________

• If A and B are independent, then Pr(A|B) = ______________

and Pr(A B) = ____________

• (Binomial Experiments): The probability of getting k successes in n

independent Bernoulli trials is:

_____________________________________


ece 450 lecture 2 - csun.edudvanalp/ece 450/ece 450 lectures/ece... · ece 450 d. van alphen 1 ece...

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