CIS 2033 based onDekking et al. A Modern Introduction to Probability and Statistics. 2007

Instructor Longin Jan Latecki

C3: Conditional Probability And Independence

3.1 – Conditional Probability Conditional Probability: the probability that an event will occur, given

that another event has occurred that changes the likelihood of the event

)(

),(

)(

)()|(

CP

CAP

CP

CAPCAP

Provided P(C) > 0

)()|(),()( CPCAPCAPCAP

For any events A and C:

3.2 – Multiplication Rule

Example: If event L is “person was born in a long month”, and event R is “person was born in a month with the letter ‘R’ in it”, then P(R) is affected by whether or not L has occurred. The probability that R will happen, given that L has already happened is written as: P(R|L)

Dec}. Nov, Oct, Sep, Aug, Jul, Jun, May, Apr, Mar, Feb, {Jan,

Dec}.Oct, Aug, Jul, May, Mar,{Jan, L

Dec}.Nov, Oct, Sep, Apr, Mar, Feb,{Jan, R

Dec}.Oct, Mar,{Jan, LR

12

8)(

7

4

127

124

)(

)()|(

RP

LP

LRPLRP

What is P(Rc| L)?

Show that P(A | C) + P(Ac | C) = 1.

Hence the rule P(Ac) = 1 – P(A)

also holds for conditional probabilities.

3.3 – Total Probability & Bayes Rule

The Law of Total Probability

Suppose C1, C2, … ,CM are disjoint events such that C1 U C2 U … U CM = Ω. The probability of an arbitraryevent A can be expressed as:

)()|(...)()|()()|()( 2211 MM CPCAPCPCAPCPCAPAP

)(...)()()( 21 MCAPCAPCAPAP

Or equivalently expressed as:

3.1 Your lecturer wants to walk from A to B (see the map). To do so, hefirst randomly selects one of the paths to C, D, or E. Next he selects randomly one of the possible paths at that moment (so if he first selected the path to E, he can either select the path to A or the path to F), etc. What is the probability that he will reach B after two selections?

3.1 Your lecturer wants to walk from A to B (see the map). To do so, hefirst randomly selects one of the paths to C, D, or E. Next he selects randomly one of the possible paths at that moment (so if he first selected the path to E, he can either select the path to A or the path to F), etc. What is the probability that he will reach B after two selections?

Define: B = event “point B is reached on the second step,” C = event “the path to C is chosen on the first step,” and similarly D and E.

P(B) = P(B ∩ C) + P(B ∩ D) + P(B ∩ E)= P(B | C) P(C) + P(B | D) P(D) + P(B | E) P(E)

3.3 – Total Probability & Bayes Rule

Bayes Rule:Suppose the events C1, C2, … CM are disjoint and C1 U C2 U … U CM = Ω. The conditional probability of Ci, given an arbitrary event A, can be expressed as:

)()|(...)()|()()|(

)()|()|(

)(

)()|()|(

2211 mm

iii

iii

CPCAPCPCAPCPCAP

CPCAPACP

AP

CPCAPACP

or

3.4 – IndependenceDefinition:

An event A is called independent of B if:

)()|( APBAP

That is to say that A is independent of B if the probability of A occurring is not changed by whether or not B occurs.

3.4 – Independence

Tests for Independence

To show that A and B are independent we have to prove just one of the following:

)()|( APBAP

)()|( BPABP

)()()( BPAPBAP

A and/or B can both be replaced by their complement.

3.4 – Independence

Independence of Two or More Events

Events A1, A2, …, Am are called independent if:

)()...()()...( 2121 mm APAPAPAAAP

This statement holds true if any event or events is/are replaced by their complement throughout the equation.

3.2 A fair die is thrown twice. A is the event “sum of the throws equals 4,”B is “at least one of the throws is a 3.”a. Calculate P(A|B).b. Are A and B independent events?

3.2 A fair die is thrown twice. A is the event “sum of the throws equals 4,”B is “at least one of the throws is a 3.”a. Calculate P(A|B).b. Are A and B independent events?

a. Event A has three outcomes, event B has 11 outcomes, and A ∩ B = {(1, 3), (3, 1)}. Hence we find P(B) = 11/36 and P(A ∩ B) = 2/36 so that

b. Because P(A) = 3/36 = 1/12 and this is not equal to P(A|B) = 2/11 the events A and B are dependent.

A computer program is tested by 3 independent tests. When there is an error, these tests will discover it with probabilities 0.2, 0.3, and 0.5, respectively. Suppose that the program contains an error. What is the probability that it will be found by at least one test? (Baron 2.5)

A computer program is tested by 3 independent tests. When there is an error, these tests will discover it with probabilities 0.2, 0.3, and 0.5, respectively. Suppose that the program contains an error. What is the probability that it will be found by at least one test? (Baron 2.5)

Example 2.18 from Baron(Reliability of backups). There is a 1% probability for a hard drive to crash. Therefore, it has two backups, each having a 2% probability to crash, and all three components are independent of each other. The stored information is lost only in an unfortunate situation when all three devices crash. What is the probability that the information is saved?

Example 2.18 from Baron(Reliability of backups). There is a 1% probability for a hard drive to crash. Therefore, it has two backups, each having a 2% probability to crash, and all three components are independent of each other. The stored information is lost only in an unfortunate situation when all three devices crash. What is the probability that the information is saved?

Example 2.19. from BaronSuppose that a shuttle's launch depends on three key devices that operate independently of each other and malfunction with probabilities 0.01, 0.02, and 0.02, respectively. If any of the key devices malfunctions, the launch will be postponed. Compute the probability for the shuttle to be launched on time, according to its schedule.

Example 2.19. from BaronSuppose that a shuttle's launch depends on three key devices that operate independently of each other and malfunction with probabilities 0.01, 0.02, and 0.02, respectively. If any of the key devices malfunctions, the launch will be postponed. Compute the probability for the shuttle to be launched on time, according to its schedule.