# Answer in Statistics and Probability for syed #231099

A box contains 10 transitors of which 2 are defective . A transistor is selected at

random from the box and tested until a nondefective one is chosen. Then the

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Answer in Statistics and Probability for syed #231099
Just from $13/Page expected number of transistors to be chosen is We will consider a more general situation, when the box contains “n” transitors, of which “m” are non-defective. The first non-defective transistor can be found in the steps “1, 2, … , n-m” or “n-m+1” . For all “k=1,\dots,n-m+1”, let “p_k” denotes the probability that the first non-defective transistor will be found in the k-th step. We can think that all the transistors are numbered with numbers 1, 2, …, n, and the number of each transistor is the same as the number of the step in which it will be chosen. The probability that a non-defective transistor will be found at the k-th step equals to [The number of ways to choose “m-1” numbers from the set “\{k+1,k+2,\dots,n\}” for non-defective transistors, starting from the second one]/[The number of ways to choose “m” numbers from the set “\{1,2,\dots,n\}” for all non-defective transistors], that is, (1) “p_k=\binom{n-k}{m-1}/\binom{n}{m}”. In particular, from this formula it follows that (2) “\sum\limits_{k=1}^{n-m+1}\binom{n-k}{m-1}=\sum\limits_{k=1}^{n-m+1}p_k\binom{n}{m}=\binom{n}{m}” By increasing “n” and “m” by 1, we also obtain that (3) “\sum\limits_{k=1}^{n-m+1}\binom{n+1-k}{m}=\binom{n+1}{m+1}” The expected number of transistors to be chosen is equal to (4) “\mu=\sum\limits_{k=1}^{n-m+1}kp_k = \sum\limits_{k=1}^{n-m+1}(n+1)p_k- \sum\limits_{k=1}^{n-m+1}(n+1-k)p_k” The first sum on the right hand side of eq.(4) is equal to (5) “\sum\limits_{k=1}^{n-m+1}(n+1)p_k= (n+1)\sum\limits_{k=1}^{n-m+1}p_k=n+1” Since (6) “(n+1-k)p_k=(n+1-k)\binom{n-k}{m-1}/\binom{n}{m}=” “=(n+1-k)\frac{(n-k)!}{(m-1)!(n-m-k+1)!}/\binom{n}{m}=” “=m\frac{(n-k+1)!}{m!(n-m-k+1)!}/\binom{n}{m}=m\binom{n-k+1}{m}/\binom{n}{m}”, then the second sum on the right hand side of eq.(4) is equal to (7) “\sum\limits_{k=1}^{n-m+1}(n+1-k)p_k= \sum\limits_{k=1}^{n-m+1}m\binom{n-k+1}{m}/\binom{n}{m}=” “=\sum\limits_{k=1}^{n-m+1}\binom{n-k+1}{m}\cdot m/\binom{n}{m}=m\binom{n+1}{m+1}/\binom{n}{m}=(n+1)\frac{m}{m+1}” (eq.(3) was applied). Finally, by substituting eq.(5) and (7) into eq.(4), we obtain (8) “\mu=(n+1)-(n+1)\frac{m}{m+1}=\frac{n+1}{m+1}” This is the answer to the question in the general case. If “n=10”, “m=8”, then “\mu=11/9”. Answer. “\mu=11/9” # What Will You Get? We provide professional writing services to help you score straight A’s by submitting custom written assignments that mirror your guidelines. #### Premium Quality Get result-oriented writing and never worry about grades anymore. We follow the highest quality standards to make sure that you get perfect assignments. #### Experienced Writers Our writers have experience in dealing with papers of every educational level. You can surely rely on the expertise of our qualified professionals. #### On-Time Delivery Your deadline is our threshold for success and we take it very seriously. We make sure you receive your papers before your predefined time. #### 24/7 Customer Support Someone from our customer support team is always here to respond to your questions. So, hit us up if you have got any ambiguity or concern. #### Complete Confidentiality Sit back and relax while we help you out with writing your papers. We have an ultimate policy for keeping your personal and order-related details a secret. #### Authentic Sources We assure you that your document will be thoroughly checked for plagiarism and grammatical errors as we use highly authentic and licit sources. #### Moneyback Guarantee Still reluctant about placing an order? Our 100% Moneyback Guarantee backs you up on rare occasions where you aren’t satisfied with the writing. #### Order Tracking You don’t have to wait for an update for hours; you can track the progress of your order any time you want. We share the status after each step. #### Areas of Expertise Although you can leverage our expertise for any writing task, we have a knack for creating flawless papers for the following document types. # Trusted Partner of 9650+ Students for Writing From brainstorming your paper's outline to perfecting its grammar, we perform every step carefully to make your paper worthy of A grade. ##### Preferred Writer Hire your preferred writer anytime. Simply specify if you want your preferred expert to write your paper and we’ll make that happen. ##### Grammar Check Report Get an elaborate and authentic grammar check report with your work to have the grammar goodness sealed in your document. ##### One Page Summary You can purchase this feature if you want our writers to sum up your paper in the form of a concise and well-articulated summary. ##### Plagiarism Report You don’t have to worry about plagiarism anymore. Get a plagiarism report to certify the uniqueness of your work. ## Free Features$66FREE

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