document.write( "Question 1208489: A slitter assembly contains 48 blades. Five blades are selected at random and evaluated each day for sharpness. If any dull blade is found, the assembly is replaced with a newly sharpened set of blades. \r
\n" ); document.write( "\n" ); document.write( "a. If 10 of the blades in an assembly are dull, what is the probability that the assembly is replaced the first day it is evaluated? \r
\n" ); document.write( "\n" ); document.write( "b. If 10 of the blades in an assembly are dull, what is the probability that the assembly is not replaced until the third day of evaluation? [Hint: Assume that the daily decisions are independent, and use the geometric distribution.] \r
\n" ); document.write( "\n" ); document.write( "c. Suppose that on the first day of evaluation, 2 of the blades are dull; on the second day of evaluation, 6 are dull; and on the third day of evaluation, 10 are dull. What is the probability that the assembly is not replaced until the third day of evaluation? [Hint: Assume that the daily decisions are independent. However, the probability of replacement changes every day.] \n" ); document.write( "

Algebra.Com's Answer #846932 by ikleyn(53765) $\"\"$ $\"About$

You can put this solution on YOUR website!
A slitter assembly contains 48 blades. Five blades are selected at random and evaluated each day for sharpness.
\n" ); document.write( "If any dull blade is found, the assembly is replaced with a newly sharpened set of blades.
\n" ); document.write( "a. If 10 of the blades in an assembly are dull, what is the probability that the assembly is replaced the first day it is evaluated?
\n" ); document.write( "b. If 10 of the blades in an assembly are dull, what is the probability that the assembly is not replaced until the third day of
\n" ); document.write( "evaluation? [Hint: Assume that the daily decisions are independent, and use the geometric distribution.]
\n" ); document.write( "c. Suppose that on the first day of evaluation, 2 of the blades are dull; on the second day of evaluation, 6 are dull;
\n" ); document.write( "and on the third day of evaluation, 10 are dull. What is the probability that the assembly is not replaced until the third day
\n" ); document.write( "of evaluation? [Hint: Assume that the daily decisions are independent. However, the probability of replacement changes every day.]
\n" ); document.write( "~~~~~~~~~~~~~~~~~~~~~\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "

\r\n" );
document.write( "(a)  So, what we have ?\r\n" );
document.write( "\r\n" );
document.write( "     We have an assembly of 48 blades.\r\n" );
document.write( "     10 blades of 48 blades of this assembly are dull.\r\n" );
document.write( "     We check 5 arbitrary blades of 48 (with no replacement).\r\n" );
document.write( "\r\n" );
document.write( "     What is the probability that at least in one trial we will find a dull blade ?\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     But of course, we should turn the sock inside out and to consider the COMPLEMETNTARY event\r\n" );
document.write( "     and calculate the COMPLEMENTARY probability that no one of 5 tested blade is dull.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     The probability that no one of 5 tested blude is dull is\r\n" );
document.write( "\r\n" );
document.write( "         P' =  = 0.3856  (rounded).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "          +--------------------------------------------------------------------+\r\n" );
document.write( "          |  You see here the product of 5 specific factors, that represent    |\r\n" );
document.write( "          |  probabilities for each of 5 consecutive tested blades to be dull. |\r\n" );
document.write( "          +--------------------------------------------------------------------+\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( " \r\n" );
document.write( "     We want the complementary value to it.  So, our  ANSWER  is  \r\n" );
document.write( "\r\n" );
document.write( "         P = 1 - P' = 1 -  = 1 - 0.3856 = 0.6144.\r\n" );
document.write( "\r\n" );
document.write( "At this point, the solution for part (a) is complete.\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "(b)  Question (b) asks you to find the probability that the given assembly of 48 blades will survive the day 1 and the day 2\r\n" );
document.write( "     (but it does not asks about day 3)\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     The answer is easy: the probability that this assembly will survive day 1 and will not be replaced after day 1 testing\r\n" );
document.write( "     is  0.3856, as we saw it in part (a).\r\n" );
document.write( "\r\n" );
document.write( "     So, if it survive, this assemble is not replaced and goes \"as is\" to the test of day 2. \r\n" );
document.write( "\r\n" );
document.write( "     The probability that this assembly will survive day 2 and will not be replaced after day 2 testing\r\n" );
document.write( "     is  the same value of 0.3856  (the second day testing is independent on the first day testing).\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "     Therefore, the probability that the given assembly will survive day 1 and day 2 testing is the product \r\n" );
document.write( "\r\n" );
document.write( "         0.3856 * 0.3856 = 0.1487  (rounded).    ANSWER\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "At this point, the solution for part (b) is complete.\r\n" );
document.write( "

\r
\n" ); document.write( "\n" ); document.write( "Having this method and these templates in front of you, you can easily solve part (c) on your own.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "All necessary ideas for it you just have in my solutions to parts (a) and (b).\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "As a conclusion, I want to say that the problem's formulation is unsatisfactory.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Indeed, in question (c), it asks \"What is the probability that the assembly is not replaced until the third day of evaluation\".\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "The precise (literal) meaning of this question in English is
\n" ); document.write( "\"What is the probability that the assembly is not replaced to the end of the second day of testing?\"\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "If so, then why in part (c) they talk about some details of activity at the day 3 ?\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "To me, it tells that the authors either do not know English, or intently try to confuse a reader,
\n" ); document.write( "or do not know, at all, what is accurate using words in Math problems.\r
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( "Or all three of these, simultaneously.\r
\n" ); document.write( "
\n" ); document.write( "
\n" ); document.write( "\n" ); document.write( " \n" ); document.write( "

\n" );