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Chapter 4: Q.4.12 (page 170)

There are n components lined up in a linear arrangement. Suppose that each component independently functions with probability p. What is the probability that no 2 neighboring components are both nonfunctional?

Short Answer

Expert verified

The answer isPElocalid="1646910197664" =鈭�0猢�m鈮�(n+1)/2n+m-1mpn-m(1-p)m

Step by step solution

01

Step 1:Given Information

Let Xdenotes the number of non functional components and let Edenotes the event. if no two nonfunctional components are to be constructive, then the space between the functional components must each contain at most one non functional components.

02

Step 2:Calculation

PE=鈭�m=0nP(E鈭�X=m)P(X=m),(From Bayes theorem)

=鈭�0鈮�m鈮�n+1)/2P(E鈭�X=m)P(X=m)

localid="1646910154485" =鈭�0鈮�m鈮�(n+1)/2n+m-1mnmnmpn-m(1-p)m

localid="1646910170371" =鈭�0鈮�m鈮�(n+1)/2n+m-1mpn-m(1-p)m

03

Step 3:Final Answer

The answer isPElocalid="1646910184882" =鈭�0鈮�m鈮�(n+1)/2n+m-1mpn-m(1-p)m

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Most popular questions from this chapter

When coin 1 is flipped, it lands on heads with probability .4; when coin 2 is flipped, it lands on heads with probability .7. One of these coins is randomly chosen and flipped 10 times.

(a) What is the probability that the coin lands on heads on exactly 7 of the 10 flips?

(b) Given that the first of these 10 flips lands heads, what is the conditional probability that exactly 7 of the 10 flips land on heads?

A box contains 5red and 5blue marbles. Two marbles are withdrawn randomly. If they are the same color, then you win \(1.10; if they are different colors, then you win -\)1.00. (That is, you lose $1.00.) Calculate

(a) the expected value of the amount you win;

(b) the variance of the amount you win.

Compare the Poisson approximation with the correct binomial probability for the following cases:

(a)P{X=2}whenn=8,p=.1;

(b)P{X=9}whenn=10,p=.95;

(c)P{X=0}whenn=10,p=.1;

(d) P{X=4}whenn=9,p=.2.

Each of 500 soldiers in an army company independently has a certain disease with probability 1/103. This disease will show up in a blood test, and to facilitate matters, blood samples from all 500 soldiers are pooled and tested.

(a) What is the (approximate) probability that the blood test will be positive (that is, at least one person has the disease)? Suppose now that the blood test yields a positive result.

(b) What is the probability, under this circumstance, that more than one person has the disease? Now, suppose one of the 500 people is Jones, who knows that he has the disease.

(c) What does Jones think is the probability that more than one person has the disease? Because the pooled test was positive, the authorities have decided to test each individual separately. The first i 鈭� 1 of these tests were negative, and the ith one鈥攚hich was on Jones鈥攚as positive.

(d) Given the preceding scenario, what is the probability, as a function of i, that any of the remaining people have the disease?

To determine whether they have a certain disease, 100people are to have their blood tested. However, rather than testing each individual separately, it has been decided first to place the people into groups of 10. The blood samples of the 10people in each group will be pooled and analyzed together. If the test is negative, one test will suffice for the 10people, whereas if the test is positive, each of the 10people will also be individually tested and, in all, 11tests will be made on this group. Assume that the probability that a person has the disease isrole="math" localid="1646542351988" .1 for all people, independently of one another, and compute the expected number of tests necessary for each group. (Note that we are assuming that the pooled test will be positive if at least one person in the pool has the disease.)
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