CIS 2033 based on Dekking et al. A Modern Introduction to Probability and Statistics 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 Provided P(C) > 0 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) What is P(R c | L)?

Show that P(A | C) + P(A c | C) = 1. Hence the rule P(A c ) = 1 – P(A) also holds for conditional probabilities.

3.3 – Total Probability & Bayes Rule The Law of Total Probability Suppose C 1, C 2, …,C M are disjoint events such that C 1 U C 2 U … U C M = Ω. The probability of an arbitrary event A can be expressed as: Or equivalently expressed as:

3.1 Your lecturer wants to walk from A to B (see the map). To do so, he first 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, he first 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 C 1, C 2, … C M are disjoint and C 1 U C 2 U … U C M = Ω. The conditional probability of C i, given an arbitrary event A, can be expressed as: or

3.4 – Independence Definition: An event A is called independent of B if: 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: A and/or B can both be replaced by their complement.

3.4 – Independence Independence of Two or More Events Events A 1, A 2, …, A m are called independent if: 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)

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 from Baron Suppose 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 from Baron Suppose 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.