CHAPTER 6 NOTES – BASIC PROBABILITY
· Two events are DISJOINT (mutually exclusive) if they have no outcomes in common.
If A and B are disjoint, then P(A or B) = P(A) + P(B)
Ex 1: One card is drawn from a deck of cards.
Let A = the card is a club B = the card is a spade
What is the probability of drawing a club or a spade?
Ex 2: In a scratch off lottery game, the probability of winning a car is .003, and the probability of winning a boat is .005. If these events are mutually exclusive, what is the probability of winning a car or a boat?
· Two events are considered INDEPENDENT if the occurrence of one event does not depend on the occurrence of the other.
If A and B are independent, then P(A and B) = P(A)P(B)
Two events A and B are independent if P(B|A) = P(B)
Ex 3: At Muhlenberg College, 55% of students are female and 45% are male. Also, 70% of the students are registered Republican, while 25% are registered Democrat. What is the probability that a student chosen at random is female and registered Republican, if these events are independent?
Ex 4: The table below displays the results of a Chamber of Commerce survey involving shoppers at a mall.
Gender / Items Shopping ForClothing / Shoes / Other / Total
Male / 75 / 25 / 150 / 250
Female / 350 / 230 / 170 / 750
Total / 425 / 255 / 320 / 1000
Are the events “female” and “shopping for shoes” independent?
INDEPENDENCE VS MUTUALLY EXCLUSIVE. Venn diagrams and multi-stage events
Ex 1: Consider the population of the Poconos. Let A be the event that a given resident enjoys hunting, and B be the event that a given resident enjoys fishing. P(A) = .45, P(B) = .60, and P(A and B) = .2. Use a Venn diagram to display this information, and answer the following.
A resident of the Poconos is chosen at random, what is…
P ( Ac )? P ( A and Bc )? P ( Ac and Bc )?
Are A and B disjoint? Why?
Are A and B independent? Why?
What is P (A or B)?
GENERAL ADDITION RULE:
Ex 2: Seventy-five percent of people who purchase hair dryers are women. Of these women purchasers or hair dryers, 30 percent are over 50 years old. What is the probability that a randomly selected hair dryer purchaser is a woman over 50 years old?
CONDITIONAL PROBABILITY: or
TREE DIAGRAMS AND CONDITIONAL PROBABILITY
Ex 3: An insurance agent knows that 70 percent of her customers carry adequate collision coverage. She also knows that of those who carry adequate coverage, 5 percent have been involved in accidents, and of those who do not carry adequate coverage, 12 percent have been involved in accidents. If one of her clients gets involved in an auto accident, then what is the probability that the client does not have adequate coverage?
PROBABILITY RULES PROVIDED ON THE EXAM:
a.
b.
Examples:
1. Which of the following statements about any two events A and B is true?
a. implies that A and B are independent.
b. implies events A and B are mutually exclusive.
c. implies events A and B are independent.
d. implies events A and B are mutually exclusive.
e. implies A and B are equally likely events.
2. If P(A) = .2, and P(B) = .1, what is , if A and B are independent?
a. .02 b. .28 c. .30 d. .32
e. There is insufficient information to answer this question
3. Given the probabilities P(A) = .4 and , what is the probability P(B) if A and B are mutually exclusive? If A and B are independent?
a. .2, .4 b. .2, .33 c. .33, .2 d. .6, .33 e. .6, .4
BASIC PROBABILITY QUESTIONS
1. The probability that a car will skid on a bridge on a rainy day is 0.75. Today, the weather station announced that there is a 20 percent chance of rain. What is the probability that it will rain today and the car will skid off the bridge?
a. 0.0300
b. 0.0375
c. 0.1500
d. 0.3000
e. 0.9500
2. The probability that Ted will enroll in an English class is 1/3. If he does enroll in an English class, the probability that he would enroll in a mathematics class is 1/5. What is the probability he enrolls in both classes?
a. 1/15
b. 2/15
c. 7/15
d. 3/5
e. 13/15
3. An automobile service station performs only oil changes and tire replacements. Eighty percent of its customers request an oil change. Of those who request an oil change, only 20 percent request a tire replacement. What is the probability that the next customer will request both an oil change and a tire replacement?
a. 0.16
b. 0.20
c. 0.25
d. 0.80
e. 0.85
4. Which of the following statements about any two events A and B is true?
a. implies that A and B are independent.
b. implies events A and B are mutually exclusive.
c. implies events A and B are independent.
d. implies events A and B are mutually exclusive.
e. implies A and B are equally likely events.
From set notation, recall that means OR, and that means AND.
Free- response question
A large security-system management company with clients in Los Angeles and San Francisco installs and monitors security systems in houses and on business premises. The system’s alarm sounds at the central monitoring location if there is any security break-in, smoke or fire at and of the client premises. Tests have indicated that in approximately 3 percent of the incidences the alarms are false, and in approximately 0.5 percent of the incidences, the system fails to sound an alarm. The records indicate that about 0.1 percent of the company’s clients have true break-in/fire/smoke incidents.
a. Suppose a client is selected at random. What is the probability that the alarm will sound at the client’s premises?
b. Suppose the alarm is sounded at a client’s premises. What is the probability that there was no break-in/smoke/ or fire incident?