Week 11 - Probability Sum and Product Rule.pdf

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Department
Computer Science
Course
CSCA67H3
Professor
Anna Bretscher
Semester
Fall

Description
Sum and Product Rules Exercise. Consider tossing a coin five times. What is the proba- bility of getting the same result on the first two tosses or the last two tosses? Solution. Let E be the event that the first two tosses are the same and F be the event that the last two tosses are the same. Let n(E) be the number of ways event E can occur. n(E) = n(F) = n(S) = Q. Is there any overlap in E and F? I.e., is E \ F = ;? A. n(E and F) = n(E \ F) = = 1 Now we can calculate the probability P(E or F): P(E or F) = = = = Theorem (The Sum Rule) If E and F are events in an experi- ment then the probability that E or F occurs is given by: P(E or F) = P(E) + P(F) ▯ P(E and F) Example. What is the probability when a pair of dice are rolled that at least one die shows a 5 or the dice sum to 8? Solution. 2 Exercise. Given a bag of 3 red marbles, 5 black marbles and 8 green marbles select one marble and then a second. What is the probability that both are red? Solution. Q. What is the probability of the first marble being red? A. Q. What is the probability of the second marble being red? A. Q. Probability of both marbles being red? A. When the probability of an event E depends on a previous event F happening we denote this probability as P(EjF). 3 Theorem (The Product Rule). If E and F are two events in an experiment then the probability that both E and F occur is: (▯) P(E and F) = P(F) ▯ P(EjF) = P(E) ▯ P(FjE) Q. What does it mean if P(EjF) = P(E)? A. Q. Therefore, if E and F are two independent events, what is (▯)? A. Example. Suppose there is a noisy communication channel in which either a 0 or a 1 is sent with the following probabilities: ▯ Probability a 0 is sent is 0.4. ▯ Probability a 1 is sent is 0.6. ▯ Probability that due to noise, a 0 is changed to a 1 during transmission is 0.2. ▯ Probability that due to noise, a 1 is changed to a 0 during transmission is 0.1. Suppose that a 1 is received. What is the probability that a 1 was sent? 4 Let A denote that a 1 was received and B denote the event that a 1 was sent. Q. What is the probability that we are solving for? A. How can we solve for this? The Product Rule says that: P(A and B) = P(B) ▯ P(AjB) = P(A)P(BjA) Therefore: P(B
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