Untitled 1
assignment a
1. an electrical unit consists of four components each of which is subject to failure. suppose that, at a specified time, we observe this unit to determine which components are working and which have failed. write the sample space associated with this random experiment.
let w denote working and f denote failed. the sample space would be
2. suppose the sample space is 
. let 
, 
, and 
. determine the sets 
here 
thus,
so,
3. let 
, 
the set of natural numbers divisible by 3, and 
the set of natural numbers divisible by 
. what is the set 
? what is the set 
?
and
so,
and
4. in a certain residential suburb 60% of all households subscribe to the metropolitan newspaper, 77% subscribe to the local paper, and 44% to both newspapers. what proportion of households subscribe to exactly one of the two newspapers?
out of 100 households, 44 subscribe to both papers. thus, 
subscribe only to the metropolitan paper and 
subscribe only to the local paper. hence, 
subscribe to exactly one of the two newspapers.
5. there are nine different locations in which a picture can be added to a text. if four different pictures are to be placed in the text, how many different designs are there?
since the picture are different, this is a permutation problem. so, the number of different designs are 
6. in how many ways a sample containing 6 non-defective and 2 defective part can be chosen from a group containing 22 non-defective and 7 defective parts?
there are
ways of choosing six non-defective parts from the group of twenty-two non-defective parts and 
ways of choosing two defective parts from the group of seven defective parts. so, the total number of choices (by multiplication principle) would be
7. how many barcodes can be formed using five a’s, four b’s and seven c’s
here we use the principle of similar permutations with 
and 
. of course,
so the required number is
8. suppose the number of ways of selecting 3 items from a group when order is important is n. what would this number be if the order was not important?
thus, if 
and 
, the number would be
assignment b:
1. two fair, distinct dice are rolled. what is the probability that the first dice comes up 1 given that the sum on the two dice is 5?
let a be the event the first die is 1, and let bbe the event the second die is 5. we want,
use this information for problems 2, 3, and 4: the following data on the marital status of 1000 u.s. adults was found in current population reports:
2.
; 
find 
according to this sample, 60.3% were married, 48% were male, and 5.2% were divorced females.
3. (a) 
(b) 
(c) probability of being widowed given that the person is a male is 14.6%. the probability that the person is a female given that the person is divorced is 10.4%.
4. are m1 and s1 independent events? justify your answer.
; 
; 
since 
the events are dependent.
5. two fair, distinct dice (one red and one green) are rolled. let a be the event the red die comes up even and b be the event the sum on the two dice is even. are a, b independent events?
by listing the elements in each event we see that 
;
and 
thus, the events are independent.
6. for the 107th congress, 18.7% of the members were senators and 50% of the senators were democrats. using the multiplication rule, determine the probability that a randomly selected member of the 107th
let d = event the member selected is a democrat and s = event the member selected is a senator
we want 
by the multiplication rule, 
7. according to the current population reports, 52% of u.s. adults are women. opinion dynamics poll published in usa today shows that 33% of u.s. women and 54% of u.s. men believe in aliens. what percentage of u.s. adults believe in aliens?
let a= event the adult selected believes in aliens and m = event the adult selected is a man. by law of total probability
; 
8. according to the american lung association 7% of the population has lung disease. of the people having lung disease 90% are smokers. of the people not having lung disease 20% are smokers. what are the chances that a smoker has lung disease?
let s= event the person selected at random is a smoker and l =event the selected person has lung disease. we want to compute 
. by bayes’ theorem, 
assignment c
use the following information to answer problems 1, 2, and 3. suppose two fair, distinct dice are rolled. let x be the random variable defined as the sum of the numbers on the two dice.
1. a) determine the probability mass function f(x).
b. determine the cumulative distribution function f(x) this is a non-decreasing function defined over all 
, and bounded between 0 and 1.
3. find the expected value and the variance of the random variable x.
to find the expected value we make the table. thus, 
as for the variance 
, we use the formula 
thus, 
4. a manufacturer claims that only 10% of its power supply units need service during the warranty period. if this claim is true, what is the probability that of the 20 units sold 4 will need service during the warranty period?
this is a binomial distribution with n =20, p= 0.1 and x = 4. this, the required probability would be 
what is expected number? 
what is the variance? 
5. a fair die is rolled over and over until a 3 is rolled.
a. what is the probability that it will take exactly four rolls?
this is a geometric distribution with x=4 and 
. thus, the required probability is
b. what is the expected value of this random variable? interpret your answer. 
in the long run, we expect the roll the die six times to see a 3.
c. what is the variance? 
d. what is the standard deviation? 
6. an oil company conducts a geological research and concludes that an exploratory oil well has 0.15 chance of striking oil.
a. what is the probability that the third strike comes in the sixth well drilled? this is a negative binomial distribution with x=6, r=3 and p=0.15. so, the required probability is
b. determine the expected value and variance of this distribution. interpret your answers.
the expected value 
and variance 
in the long run, we expect to drill twenty holes to get three strikes.
7. a bag contains twenty marbles: six red and fourteen black. five marbles are drawn without replacement. what is the probability that exactly two are red?
this is a hypergeometric distribution. so, the required probability is 
8. if electricity power failures occur according to a poisson distribution with an average of three failures every twenty weeks, calculate the probability that there will not be more than one failure during a particular week. 
per twenty weeks 
per week
we are looking for 
. 