Discrete Math for CS CMPSC 360 LECTURE 32 Last time: Review. Today: Combining events. Conditional probability. CMPSC 360 11/15/2018

Monty Hall problem 1970 game show hosted by Monty Hall. A contestant is shown 3 doors: behind one is a prize and behind the other two are two goats. The contestant picks a door, but does not open it. Then one of the other two doors is opened to reveal a goat. The contestant is given the option of sticking with his choice or picking the other unopened door. He wins the prize iff he picks the door with the prize. Should he switch? 11/15/2018 Picture source: http://en.wikipedia.org/wiki/Monty_Hall_problem

I-clicker question (frequency: BC) When do you have a higher probability of winning? If you stick with your first choice. If you switch. Both probabilities are the same. 11/15/2018 Picture source: http://en.wikipedia.org/wiki/Monty_Hall_problem

How to solve a probability problem Figure out: the sample space (the set of possible outcomes). the probability of each sample point (outcome). the event (subset of sample space) you are interested in. Compute the probability of the event by adding up probabilities of sample points in it. 50+33+20-16-10-6+3=103-32=3=68 11/15/2018

I-clicker question (frequency: BC) A dart is thrown blindly at the target shown on the board. The probability that the dart lands in one of the three concentric regions is proportional to the area of the region. The radii of the circles are 1,2, and 3 units. Let 𝑝 𝑖 be the probability that the dart lands in region 𝑖. 𝑝 1 = 𝑝 2 = 𝑝 3 𝑝 1 < 𝑝 2 < 𝑝 3 𝑝 1 > 𝑝 2 > 𝑝 3 None of the above. \pi,3\pi, 5\pi, 11/15/2018 Picture source: http://en.wikipedia.org/wiki/Monty_Hall_problem

Combining events Union of events: 𝐴∪𝐵 Intersection of events: 𝐴∩𝐵 A or B occurs (or both) Intersection of events: 𝐴∩𝐵 Both A and B occur Difference of events: 𝐴−𝐵 A occurs, but B does not Complement of an event: 𝐴 A does not accur. 11/15/2018

Inclusion-Exclusion Recall: for any two sets A and B, 𝐴∪𝐵 = 𝐴 + 𝐵 − 𝐴∩𝐵 . For any two sets A and B, Pr 𝐴∪𝐵 = Pr 𝐴 + Pr 𝐵 −Pr⁡[𝐴∩𝐵]. 11/15/2018

Conditional probability A= event that the student misses school bus B= event that the student’s alarm malfunctions What is the probability that the student misses school bus given that her alarm clock malfunctions? 11/15/2018

Example: two dice Event A: the numbers on the dice sum to 8. Event B: the numbers on the dice are both even. What is the probability of each event? What is the probability of A given that B occurred, Pr⁡[𝐴|𝐵]? 11/15/2018

Conditional probability: definition Let A and B be events, and suppose that Pr 𝐵 ≠0. The conditional probability Pr 𝐴 𝐵 , the probability of A given B, is Pr 𝐴 𝐵 = Pr 𝐴∩𝐵 Pr⁡[𝐵] . 11/15/2018