Math 3339 Section 21155 - SR 117 - MW 1-2:30pm Bekki George: bekki@math.uh.edu University of Houston Sections 3.3-3.4 Bekki George (UH) Math 3339 Sections 3.3-3.4 1 / 12
Office Hours: Mondays 11am - 12:30pm, Tuesdays 3-4pm (also available by appointment) Office: 206 PGH Course webpage: www.casa.uh.edu Bekki George (UH) Math 3339 Sections 3.3-3.4 2 / 12
Rules for Probability Measures Axioms, Interpretations, and Properties of Probability Given a sample space, our goal is to assign to each event in the sample space a probability which will give us the chance that the event occurs. Axioms: 1 For any event E, 0 P (E) 1 2 P (Ω) = 1 3 If E 1, E 2, E 3,... is an infinite collection of disjoint events, then P (E 1 E 2 E 3...) = P (E i ) i=1 Bekki George (UH) Math 3339 Sections 3.3-3.4 3 / 12
Rules for Probability Measures More Properites: 4 P ( ) = 0 5 For any event E, P (E) + P (E ) = 1 6 For any two events E and F, P (E F ) = P (E) + P (F ) P (E F ) 7 If E F,then P (E) P (F ) Bekki George (UH) Math 3339 Sections 3.3-3.4 4 / 12
Equally Likely Outcomes In an experiment with N equally likely outcomes, what is the probability of each outcome? Ex: What is the probability of rolling a 2 on a fair die? Bekki George (UH) Math 3339 Sections 3.3-3.4 5 / 12
Using Probability Rules Ex: In a certain residential suburb, 60% of all households get internet service from the local cable company, 80% get television service from that company, and 50% get both services from that company. If a household is randomly selected, what is the probability that it gets at least one of these services from the company? Bekki George (UH) Math 3339 Sections 3.3-3.4 6 / 12
Using Probability Rules Ex: During off-peak hours a commuter train has five cars. Suppose a commuter is twice as likely to select the middle car (#3) as to select either adjacent car (#2 or #4), and is twice as likely to select either adjacent car as to select either end car (#1 or #5). Determine the probability that one of the three middle cars is selected. Bekki George (UH) Math 3339 Sections 3.3-3.4 7 / 12
Examples with Counting Rules Ex: A hand of 5-card draw poker is a simple random sample from the standard deck of 52 cards. What is the probability that a 5-card draw hand contains the ace of hearts? Bekki George (UH) Math 3339 Sections 3.3-3.4 8 / 12
Examples with Counting Rules Ex: A university warehouse has received a shipment of 25 printers, of which 10 are laser printers and 15 are inkjet models. If 6 of these 25 are selected at random to be checked by a particular technician, what is the probability that exactly 3 of those selected are laser printers? What is the probability that at least 3 laser printers are selected? What is the probability that at least one laser printer is selected? Bekki George (UH) Math 3339 Sections 3.3-3.4 9 / 12
More Examples Among 6 electrical components exactly one is known not to function properly. If 4 components are randomly selected, find the probability that all selected components function properly. find the probability that exactly one does not function properly. find the probability that at least one does not function properly. Bekki George (UH) Math 3339 Sections 3.3-3.4 10 / 12
More Examples The probability that a randomly selected person has high blood pressure (the event H) is P(H) = 0.2 and the probability that a randomly selected person is a runner (the event R) is P(R) = 0.3. The probability that a randomly selected person has high blood pressure and is a runner is 0.1. Find the probability that a randomly selected person has high blood pressure and is not a runner. Bekki George (UH) Math 3339 Sections 3.3-3.4 11 / 12
More Examples In a shipment of 66 vials, only 17 do not have hairline cracks. If you randomly select 3 vials from the shipment, what is the probability that two of the vials have hairline cracks? Bekki George (UH) Math 3339 Sections 3.3-3.4 12 / 12