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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 w8MgooHMhmQAAAAASUVORK5CYII=.  let Ah3KH1ZdVC6IAAAAAElFTkSuQmCC, P7QsOtpxon0ScHGICU69ueneJYv4B4EfzfMfJs+B, and GVqf0R6ASYXx9u0HKKzAAAAABJRU5ErkJggg==.  determine the sets rJ4EMhaCNY2UvgAAAABJRU5ErkJggg==


3. let Zu4Tlr8b3J6NIswAAAAASUVORK5CYII=, 7TVxAAAAQElEQVQoU2NgoAAIc3Gi6xZgY0cTEuIQ the set of natural numbers divisible by 3, and 7TVxAAAATUlEQVQoU2NgIBuIMDIyMvGjaBfn4WcQ the set of natural numbers divisible by 7TVxAAAAMElEQVQYV2NgwAYEOBgZmbkZBDh5QLIw.  what is the set bx7TwTJXwJa2mL+Rsgppwxf8T9yBSnjBuEKNPqSA?  what is the set hGHjPoBuFDeVx1AAAAAElFTkSuQmCC?PBR34Bl2PLPcNwRnLAAAAAElFTkSuQmCCandg10FB+vSr06ZwAAAABJRU5ErkJggg==


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, xpXd2LCdSmrQrKgAAAABJRU5ErkJggg== subscribe only to the metropolitan paper and 8V93fAwa7nN2iRUAAAAASUVORK5CYII= subscribe only to the local paper.  hence, esDjmIsCnSayu8AAAAASUVORK5CYII= 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  VYSLKZ7sC4kWKk8F5JP745ZS+7pTjH55nuvDBupL

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 areLi1Xva4lQjPX5GPtyACX4DM0w5SzTM5CfKRnVSuvways of choosing six non-defective parts from the group of twenty-two non-defective  parts and  CAaWO3F5YWU+rP8eMaSnYI38H8LlZ8l7vEAAAAASways of choosing two defective parts from the group of seven defective  parts.  so, the total number of choices (by multiplication principle) would beeQE1eswibwgAAAABJRU5ErkJggg==

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 CTIBfUwAAAABJRU5ErkJggg== and YzJR0aR71szaYNHjMyTHHPAszIsb6fcB03GbHZ6S.  of course,nCm3k83uNTI7EYoBGrhOjm9oLM+PbgIPDyltN30E        so the required number isPYj8CUiHf6oAhDRkLnTgqbOUTgJZfRbANmdb8Kdv

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?

  +Rj+Y8+SAW+AVryVRiuI0vOAAAAAElFTkSuQmCC thus, if EMNWvnOPUbW0Owbs5WGcDa3ZkIXVcLjkvqe9Zrxs and r4Wx2QPQIikGHiNl27adhygoTMsypw0+KUG6KVRR, the number would be3vCTqADS1jIsnsAAAAAElFTkSuQmCC

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,wH24puN948rTAAAAAASUVORK5CYII=

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.zcCGZb7gDKpxj5Krff9GIQ+lMYBl5w0fAVF7gIQJ;   7Ao96U9TGuIytAAAAAElFTkSuQmCC find AanbJCgU4TwK0IFKKTM5OxfhPFDpKMcZ6QxfM5Nw

according to this sample, 60.3% were married, 48% were male, and 5.2% were divorced females.

3.   (a) AeeA1uZbeHHdAAAAAElFTkSuQmCC

(b) TdlAcK520TuSdijMfSUr7+jmGy5bUpibkJUbnuHH

(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.

7zemBDu7QpOBUAAAAABJRU5ErkJggg==s9waB7MrrS9D7882cyutL0rcH5GUxDIZlfa3hX4h; Zpqe4eE2AAAAAElFTkSuQmCC

cDz7RJVpyuwAAAAASUVORK5CYII=since 2lmgAAAABJRU5ErkJggg== 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 biCuPpEK5n1tkaSjcOyrUWK30lTl2UfvigrGGmwV

;5KeUX1erEMUS9sbyBCkcw8fgnlSG2sbcrkPUFvFo and DznJxqPU7616jqeNg4Enudbo62B+u8Uyh6CHt6AAthus, 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 z9+ROZZhlbWriyAQAAAABJRU5ErkJggg==  by the multiplication rule, 6ASRO6LObQe1wdx0DIVZTBAbDhfgNwGd1YfwD9y8

 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 KDGxNrUr1qjePhVcJQciGmmiGWutwqQ9EZLNKrI8. by bayes’ theorem,  gcwQmGCdDqnhwAAAABJRU5ErkJggg==

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 2wECAwECAwECAwECAwECAwECAwECAwECAwECAwEC, 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,  nWW0q0S8xb73jpMtalv3XMAyAj9CYVFmM+HToUDJ

as for the variance QiSNiwdZcAZgKW5aJRuHRSSYACkGQywRdIhZKVVD, we use the formula EFmvsscgmq+yyzDbrrK0xWIGDoc9WS2ILMTSQAQ3 thus, mJKvPOSPQwQwPrWrrylh4AKqLEPMv8h1lVXhkArN

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 PrrX3EPmyLrJJNSiuBg77U8V5ZdSKvSzexhG0UmM

what is expected number?  crNDD6eKmIBMIAixcRbm5psU1IiEgogjEHOR6dfm

what is the variance?  VzHDT3bLQhZjzhxj8mn7IGJSYvffQjT8MhtRBJv3

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 2wECAwECAwECAwECAwECAwECAwECAwECAwECAwEC.  thus, the required probability is+edEgAPO+266cxgkBItOvDhwGFMQ7GIAz84mk0Sw

b. what is the expected value of this random variable?  interpret your answer.  wPBXEKEQvk4mphnJJ46cRg6mTupSeNRlgIy97r1w in the long run, we expect the roll the die six times to see a 3.

c. what is the variance?  swQBADs=

d. what is the standard deviation? 3XIBQfAVIVuQPIzNcRRBqZs41biqVGu3TkjTt6tF

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 ismqCeOOUI6PGChHcDEflnPMSBAA7

b. determine the expected value and variance of this distribution. interpret your answers.

the expected value ssCAgEYxjuvrn76xfbNQQw4OygIADs= and variance 4Ba4SQQAAOw==

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 YlsJf5JjHkqBoIGxyaOagB+UbQJeGbjoYN3RgVuC

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. 2wECAwECAwECAwECAwECAwECAwECAwECAwECAwEC per twenty weeks 2wWNICCOZGmeKFlpKYodQcGanSONV6DHc0tuAskn per week

we are looking for Tv9++HgQAOw==UNSAVhbNykc3StKS1iGhHgUp7S6qUpO69KUmcTTQ