Probability
Suppose an experiment (such as dropping a thumbtack, tossing a coin, or selecting a candy from a bag of M&M’s) is
repeated many times. The relative frequency associated with an outcome (such as the tack landing with the point upward,
or heads rather than tails in the coin toss, or selection of a yellow candy) will tend toward some number between 0 and 1
(inclusive) as the number of repetitions increases. The long-term relative frequency approaches the probability of the
outcome.
A chance experiment is any activity or situation in which there is uncertainty concerning which of two or more possible
outcomes will result.
The probability of an outcome is interpreted as the long-run proportion of the time that the outcome would occur, if the
experiment were repeated indefinitely. That is, probability is long-term relative frequency.
Examples:
P(Head) = 1/2
P(2) = 1/6
Sample Space is the set of all possible outcomes.
Examples:
S= {H,T}
S= {1,2,3,4,5,6}
Rules for Probability
1.
2.
The probability of each outcome must be a number between 0 and 1, inclusive.
The probabilities of all of the outcomes in a given sample space must add up to one.
Calculating the probability for an event
An event is a specified set of possible outcomes in the sample space.
A – An even number shows
A= {2,4,6}
The probability of an event equals the sum of the probabilities associated with the outcomes that constitute that event.
P(A)=1/6+1/6+1/6=1/2
Mutually Exclusive Events
Two Events A and B are mutually exclusive if when an experiment is conducted a single time, the occurrence of one
event excludes the possibility of the occurrence of the other event.
A – A 1 shows on the die
B – A 6 shows on the die
P(A or B)= 1/6 + 1/6=2/6=1/3
In general,
P(A or B) = P(A) + P(B) – P(A and B)
If events A and B are mutually exclusive, P(A and B) = 0.
Conditional Probability:
Let E and F be two events with P(F)>0. The conditional probability of the event E given that the event F has
occurred, denoted by P(E|F), is
P(E|F)=
P( E and F )
P(F)
Independent Events
Two events E and F are said to be independent if the probability of E occurring is not affected by event F
having occurred or vice versa.
Two events E and F are said to be independent if
P(E|F)=P(E)
If E and F are not independent, they are said to be dependent events.
In-Class Exercise: Probability
1. The table below gives data on the age and marital status of adult American women.
Table Age and Marital Status of Women (thousands of women)
Age
18 to 24
25 to 64
65 and over
Total
Married
3,046
48,116
7,767
58,929
Never Married
9,289
9,252
768
19,309
Widowed
19
2,425
8,636
11,080
Divorced
260
8,916
1,091
10,267
12,614
68,709
18,262
99,585
Total
One of the adult women is chosen at random. What is the probability that the woman is
a.
b.
c.
d.
Widowed
Married and between 25 and 64 years old
Married or between 25 and 64 years old
Married given the information that she is between 25 and 64 years old.
2. A mail-order house employs three stock clerks, P, Q, and R, who pull items from shelves and assemble them for subsequent
verification and packaging. P makes a mistake in an order (gets a wrong item or the wrong quantity) one time in a hundred, Q makes a
mistake in an order 5 times in a hundred, and R makes a mistake in an order 3 times in a hundred. Off all the orders delivered for
verification, P, Q, and R fill, respectively, 30, 40, and 30 percent. If a mistake is found in a particular order, what is the probability that
it was filled by Q?
3. A fair coin is to be flipped four times. Find the following probabilities of observing the following:
a.
b.
c.
d.
no heads
exactly one head
exactly two heads
at least one tail
4. A personnel manager has cross-classified the 400 employees of a firm according to their record of absenteeism last year and
according to whether or not they were smokers, as shown in the accompanying table. One of these employees is selected at random.
Number of Days Absent
Smoker
Nonsmoker
Less than 10
34
260
10 or more
78
28
a. Are the events "nonsmoker" and "absent less than 10 days" mutually exclusive? Explain.
b. Determine whether an employee’s being absent for 10 or more days last year was independent of the employee’s being a smoker?
Show work!
5. Two fair dice are tossed and the following events are defined:
A: {Sum of the numbers showing is odd}
B: {Sum of the numbers showing is 8, 9, 11, or 12}
a.
b.
Are events A and B independent? Why?
Are events A and B mutually exclusive? Why?
6. An important component of a small computer is a microchip, 20% of which are purchased from supplier A1, 50% from supplier A2,
and 30% from supplier A3. Suppose it is known that the proportions of defectives produced by the three suppliers are .001, .0003, and
.0007, respectively. If a single computer microchip failure is observed, which supplier is most likely responsible?
7. If one card is drawn at random from a deck of cards, find the probability of getting
a.
An ace
b.
A diamond
c.
An ace of diamonds
d.
A 4 or a 6
e.
A and a 6
f. A 4 or a club
g. A 6 or a spade
h. A heart or a club
i. A red queen
8. In a college of 1000 students and 100 faculty, 10% of the faculty are Democrats and 90% are Republicans, whereas among the
students these percentages are reversed. A member of the college is drawn at random. Determine the following probabilities.
a. P(Student)
b. P(Democrat and Faculty)
c. P(Student or Republican)
d. P(Student | Republican)
9. Use the following table to answer the questions below.
Headache
No Headache
Total
Seldane
49
732
781
Placebo
49
616
665
Control Group
24
602
626
Total
122
1950
2072
a. If one of the 2072 subjects is randomly selected, find the probability of getting someone who used a placebo or was in the control
group.
b. If one of the 2072 subjects is randomly selected, find the probability of getting someone who used Seldane or did not experience a
headache.
10. For a three-question multiple-choice pop quiz, a student is totally unprepared and randomly guesses the answer to each question. If
each question has four options, then the probability of selecting the correct answer for any given question is 1/4, or .25. With guessing,
the response on one question is not influenced by the response on another question. Thus, whether one question is answered correctly
is independent of whether or not another question is answered correctly. Find the probability the student passes, answering at least two
questions correctly.