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An addictions counsclor records information about thc numbcr of calls for hclp that arc reccived at hcr trcatment centcr's hotlinc: Shc ratcs cach wcck's ...

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An addictions counsclor records information about thc numbcr of calls for hclp that arc reccived at hcr trcatment centcr's hotlinc: Shc ratcs cach wcck's call volumc as hcavy or light and hcr rccords show that if onc wcck's call volumc was hcavy: thcre $ a 60% chancc thc ncxt wcck's volumc will bc hcavy too. Ifa givcn wcck's call volumc was light; thcrc 5 a 50% chancc thc next wcck's volumc will be heavy: If the weckly call volume can be modeled as a Markov Chain; t

An addictions counsclor records information about thc numbcr of calls for hclp that arc reccived at hcr trcatment centcr's hotlinc: Shc ratcs cach wcck's call volumc as hcavy or light and hcr rccords show that if onc wcck's call volumc was hcavy: thcre $ a 60% chancc thc ncxt wcck's volumc will bc hcavy too. Ifa givcn wcck's call volumc was light; thcrc 5 a 50% chancc thc next wcck's volumc will be heavy: If the weckly call volume can be modeled as a Markov Chain; then what are the chances to have three consecutive weeks of heavy call volume at her treatment center? None of the Otners are correct 56% 20% 21.6%6 55,6%6 36%



Answers

Therapy: Alcohol Recovery The Eastmore Program is a special program to help alcoholics. In the Eastmore Program, an alcoholic lives at home but undergoes a two-phase treatment plan. Phase I is an intensive group-therapy program lasting 10 weeks. Phase II is a long-term counseling program lasting 1 year. Eastmore Programs are located in most major cities, and past data gave the following information based on percentages of success and failure collected over a long period of time: The probability that a client will have a relapse in phase I is $0.27 ;$ the probability that a client will have a relapse in phase II is $0.23 .$ However, if a client did not have a relapse in phase I, then the probability that this client will not have a relapse in phase II is 0.95. If a client did have a relapse in phase I, then the probability that this client will have a relapse in phase II is 0.70. Let $A$ be the event that a client has a relapse in phase I and $B$ be the event that a client has a relapse in phase II. Let $C$ be the event that a client has no relapse in phase I and $D$ be the event that a client has no relapse in phase II. Compute the following: (a) $P(A), P(B), P(C),$ and $P(D)$ (b) $P(B | A)$ and $P(D | C)$ (c) $P(A \text { and } B)$ and $P(C \text { and } D$ ) (d) $P(A \text { or } B)$ (e) What is the probability that a client will go through both phase I and phase II without a relapse? (f) What is the probability that a client will have a relapse in both phase I and phase II? (g) What is the probability that a client will have a relapse in either phase I or phase I??

All right. So this time we are selecting three people at a time and we're wondering what's the probability that they all had false positive test results? So um are false positives are right here. So we'll start at 24 out of 300. Okay. That's our first person having a false positive. Then our second person having a false positive would be the 23 out of 300 that remain. But actually we're not out of 300 because we took one out. So it's 2 99 And then we'll take another one out to get 22 out of 298. Now this is going to come out to be a probability of 0.000454 Which is less than 5% and therefore unlikely. Yeah.

In this problem, we are going to determine the probabilities of certain events. A table is given and two events are given. Is that the offender has 10 or more years of education and B is the event that the offender is convicted within two years after completion of treatment. Now in the first problem, we need to determine the value of p. of E. So that means we need to find the probability that the offender has 10 or more years of education. Now from the table, the total And with the offender has generally year 10 or more years of education is 0.40. So that's the Numerator in the probability and the total is 1.00 from the table. So the probability is 0.40 by 1.00 which is just equals to 0.40. And the second problem, we need to find the value of PMP Which is the probability that the offender is convicted within two years after completion of treatment. And from the table we can see that the numerator will need to be the total for the N. B. Which is he goes to 0.37 And the total number of outcomes. The denominator would be like the previous case 1.00. So that is 0.37 x one which is just equals to 0.37. Next we need to find the value of P. E. Intersection B. Once again the total is 1.00. And from the table b. Of a intersection be the total for a intersection. B means the offender has 10 memorials of education and has also been convicted within two years after completion of treatment. So the total in this case will be 0.10. Which is obtained from the table. So To be 0.10 or 0.1 next we need to find the value of P. E. Union. Be now be of a union, be can be obtained using the addition rule. And the additional rule says the p of a union B. Is supposed to be of A plus P. Of B minus B of a intersection B. So we just found the values of P. A P P and P a intersection B. So we can substitute those values 0.400 point 37 And this would be minus 0.1. So the value for this will be equals to 0.67. Mhm. Next we need to find the value of P. Of a bar. So that's the complementary way. So this is equal to one minus B. Of a. So that's one minus the value of P. Of a 0.40 So one minus 0.40 is equal to 0.60 or 0.6. Next we need to find the value of P. Of the union rebar and that is equal to the value of one minus P. Of a union. Be and that is equals two, 1 -0.67. We just calculated developed p of a union media equals to 0.67 and 1 -0.67 is equals to 0.33. So there is a probability in this case. Next we need to find the value of P. L. E. Intersection B. Bar. So there's a complimentary intersection B. So that we want minus B. Of a intersection B. So that's 1 -0.1, which is equals to 0.9. And that is the probability in this case. Next we need to find the value of P. O. E. Given be. Now using the definition of conditional probability, this is equal to P. Of a intersection, be divided by the F B. The value of p. of a intersection B is equals to 0.1 and the value of PLB equals to 0.37. So that is equal to 10, divided by 37. And that is the probability in this case and last. We need to find the probability of being given. Uh So by the definition of conditional probability that is equal to P of the intersection, he divided by P. L. E. Now, since the intersection, any intersection B is the same thing, so this would be of a intersection be divided by P. L. E. Now the of a intersection B is equal to 0.1 B. Of a. Has already been determined to be 0.40 So that would just be 10 by 40 or one by four. And that is the probability in this case.

So 45% of people know someone who is addicted to a drug over than alcohol. So this is a binomial random variable because there are two options you ever addicted to this drug or you're not. So that means we can use this equation to find probabilities because this gives us probabilities of binomial random variables. And I think you should have this running down now for the 1st 1 buying probability of free from a sample of five. So the free pieces of information we need our five, the free and then this 45 up here five is our sample that's N X is gonna be free. That's the number we're looking for. And a probability is 45% Now. We're going to convert this to a decimal, though, so it's going to be point for five. So to find the probability of free that will be equal to the sample. Five choose free times the probability so 0.45 Tioga to the one minus point. Bert multiplied by one minus 10.45 raised to the five minus free power. And this is going to give us 0.276 Now we have a sample of 15 people, and we need to find the probability that seven of them are addicted to the drug. But you need to use the table in the back of the book and appendix B table to it gives you a table that looks something like this. I cannot post pictures directly from the books. I recreated some of it. So you're gonna scroll down to the end equals 15 section. And because our probability is 45% you're gonna be in between the 450.40 and the 0.50 So I wrote out some of the relevant section here. So the fine probability of seven Well, we go to seven. And because it's 70.45 we're just gonna take the average of these two numbers right here. So probability of seven is equal 2.177 plus 0.196 over to. And we're doing this because we're just taking the average of these two, and this gives us 0.1865 Now, these last to find the probability that it's a least seven, So we're still gonna be in this section. So to find the probability of a least seven, you're going to start with seven. So that's here and again. We need both, and you're going to go all the way down until you reach X equals 15. We'll keep going down until you get two X equals 15 and you're gonna add all of these up and divine my chill to find the average. Now, you should get 0.5 or 25 or probability of at least seven. And now, for our last part, find the probability of less than or equal to seven. So hopefully you didn't clear your calculator because if you didn't clear your calculator, all you gotta do is delete this 0.177 and this 0.196 and then same thing. Divide the rest of them by two and then one minus it. However, if he didn't do that, what you're gonna do now is starting again at seven. You're going to go up all the way to X equals zero. So you're gonna take all the cases from zero up to seven for the 70.4 in the 0.5 column Adam all up and divide by two to take the average, just like in the last one, and you should get probability of less than or equal to seven is equal 2.644 Now, if you didn't clear your calculator, just delete the 0.177 in the 0.196 I don't give you the probability of a coup least eight and then one minus it, because the probability of greater than or equal to seven is equal toe one, minus all the other probabilities. Since all probabilities at upto wine, you can save a little bit of time here if you used a calculator and we are done.

This time around, we are sampling three subjects randomly without replacement and wondering about have the probability that they all have true negative results. So it's going to be right here are 154. So the probability of the first person has, the test result is going to be 154 out of the 300 And we're not replacing. So that means that there's 153 out of 299 left from the negatives, And finally 152 Out of 298 For our negatives. And this is going to come out to be a probability of 0.134 or 13.4%,, Which is much bigger than our 5%. And therefore it is not unlikely for this to occur.


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