Review of Basic Probability Rules Complement Rule P(A) = 1 – P(Ac) Addition Rule for Mutually Exclusive Events
P(A or B) = P(A) + P(B) Multiplication Rule for Independent Events P(A and B) = P(A)*P(B)
“At Least One” Rule P(At least one) = 1 – P(none) 1
An Infinite Sample Space We observe a production line until a defective (D) item appears. The sample space now is infinite since the event may never occur. The sample space is shown below (where G denotes a good item).
Tossing a Coin
• In a group of 200 households, 124 own telephone answering machines. Select randomly one household from this group. What is the probability that this household owns a telephone answering machine?
The following table gives the two-way classification of 400 students based on sex and whether or not they work while being full-time students. Male Female
Work 120 130
Do Not Work 60 90
a. Select one student randomly from this group of 400 students. What is the probability that this student: (i) does not work (ii) is a female (iii) does not work given he is a male (iv) is a female given she works b. Are the events “male” and “do not work” mutually exclusive? Explain why or why not. c. Are the events “female” and “do not work” independent? Explain why or why not. d. What is the complementary event of the event “do not work”? What is the probability of this complementary event?
a.(i) 150 400 = .375 (ii) 220 400 = .550 (iii) 60 180 = .333 (iv) 130 250 = .520 b. The events are not mutually exclusive because they can happen together. c. P(female) = .550, P(femaledoes not work) = .600. Hence, these two events are not independent. d. The complementary event is “work” and its probability is .625
An independent research team inspects 300 batteries manufactured by two companies for being good or defective. The following table gives the two-way classification of these 300 batteries. Company A Company B
Good 140 130
Defective 10 20
a. The team selects one battery randomly from these 300 batteries. Find the probability that this battery: (i) is manufactured by company B (ii) is defective (iii) is good given that it is manufactured by company B (iv) is manufactured by company A given that it is defective b. Are the events “company A” and “defective” mutually exclusive? Explain why or why not. c. Are the events “good” and “company A” independent? Explain why or why not. d. What is the complementary event of the event “defective”? What is the probability of this complementary event?
a.(i) 150 300 = .500 (ii) 30 300 = .100 (iii) 130 150 = .867 (iv) 10 30 = .333 b. Not mutually exclusive since they can happen together c. P(good) = .900, P(goodcompany A) = .933. Hence, these events are not independent. d. The complementary event is “good” and its probability is .900
There are a total of 300 professors at a university. Of them, 75 are female and 90 are professors in the social sciences. Of the 75 females, 30 are professors in social sciences. Are the events “female” and “professor in social sciences” independent? Are they mutually exclusive? Explain why or why not?
P(Female) = 75 300 = .25, P(FemaleProfessor in social sciences) = 30 90 = .333. Hence, these events are not independent. The events are not mutually exclusive because there are 30 female professors in social sciences (joint occurrence).
no
Ch 5
3-sol
• Or D. tree • Or law of comb.
2-sol
3-sol
2-sol
• p(AB)=P(A) P(B)