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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" =∑0⩽m≤(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=∑m=0nP(E∣X=m)P(X=m),(From Bayes theorem)

=∑0≤m≤n+1)/2P(E∣X=m)P(X=m)

localid="1646910154485" =∑0≤m≤(n+1)/2n+m-1mnmnmpn-m(1-p)m

localid="1646910170371" =∑0≤m≤(n+1)/2n+m-1mpn-m(1-p)m

03

Step 3:Final Answer

The answer isPElocalid="1646910184882" =∑0≤m≤(n+1)/2n+m-1mpn-m(1-p)m

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A ball is drawn from an urn containing 3white and 3black balls. After the ball is drawn, it is replaced and another ball is drawn. This process goes on indefinitely. What is the probability that the first 4balls drawn, exactly2 are white?

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