document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1177158>Question 1177158</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A lot of 50 electrical components numbered 1 to 50 is drawn at random, one by one, and is<BR>\n" );
document.write( "divided among five customers.<BR>\n" );
document.write( "(a) Suppose that it is known that components 3, 18, 12, 26, and 46 are defective. What is<BR>\n" );
document.write( "the probability that each customer will receive one defective component?<BR>\n" );
document.write( "(b) What is the probability that one customer will have drawn five defective components?<BR>\n" );
document.write( "(c) What is the probability that two customers will receive two defective components each,<BR>\n" );
document.write( "two none, and the other one?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850509 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1987)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850509\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **Assumptions:**\r<BR>\n" );
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document.write( "* Each component is equally likely to be drawn.<BR>\n" );
document.write( "* The drawing is done without replacement (once a component is drawn, it's not put back in).\r<BR>\n" );
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document.write( "**(a) Probability that each customer receives one defective component:**\r<BR>\n" );
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document.write( "1. **Calculate the total number of ways to distribute the components:**<BR>\n" );
document.write( "   * Each of the 50 components can go to any of the 5 customers.<BR>\n" );
document.write( "   * Total ways to distribute: 5^50\r<BR>\n" );
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document.write( "2. **Calculate the number of ways to distribute one defective component to each customer:**<BR>\n" );
document.write( "   * We need to assign one defective component to each customer.<BR>\n" );
document.write( "   * Number of ways to assign defective components: 5! (5 factorial)<BR>\n" );
document.write( "   * The remaining 45 non-defective components can be distributed among the customers in 5^45 ways.\r<BR>\n" );
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document.write( "3. **Calculate the probability:**<BR>\n" );
document.write( "   * Probability = (Favorable ways) / (Total ways)<BR>\n" );
document.write( "   * Probability = (5! * 5^45) / 5^50<BR>\n" );
document.write( "   * Probability = 5! / 5^5 = 120 / 3125 = **0.0384**\r<BR>\n" );
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document.write( "**(b) Probability that one customer has all five defective components:**\r<BR>\n" );
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document.write( "1. **Choose the customer:**<BR>\n" );
document.write( "   * There are 5 ways to choose the customer who gets all the defective components.\r<BR>\n" );
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document.write( "2. **Calculate the probability for that customer:**<BR>\n" );
document.write( "   * The probability that the first defective component goes to this customer is 1/5.<BR>\n" );
document.write( "   * The probability that the second defective component also goes to this customer is 1/5 (since we're drawing without replacement).<BR>\n" );
document.write( "   * And so on for all five defective components.<BR>\n" );
document.write( "   * Probability for this customer = (1/5)^5\r<BR>\n" );
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document.write( "3. **Calculate the probability for the remaining components:**<BR>\n" );
document.write( "   * The remaining 45 components must go to the other 4 customers.<BR>\n" );
document.write( "   * Probability for remaining components = 4^45 / 5^45\r<BR>\n" );
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document.write( "4. **Calculate the overall probability:**<BR>\n" );
document.write( "   * Probability = (5 ways to choose customer) * (Probability for that customer) * (Probability for remaining components)<BR>\n" );
document.write( "   * Probability = 5 * (1/5)^5 * (4^45 / 5^45) = 4^45 / 5^49  ≈ **1.055 x 10^-6** (very small probability)\r<BR>\n" );
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document.write( "**(c) Probability that two customers receive two defective components each, two none, and the other one:**\r<BR>\n" );
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document.write( "1. **Choose the customers:**<BR>\n" );
document.write( "   * There are 5C2 = 10 ways to choose the two customers who get two defective components each.<BR>\n" );
document.write( "   * There are 3C2 = 3 ways to choose the two customers who get no defective components.\r<BR>\n" );
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document.write( "2. **Assign defective components:**<BR>\n" );
document.write( "   * We need to divide the 5 defective components into two groups of 2 and one group of 1.<BR>\n" );
document.write( "   * Number of ways to do this: 5! / (2! * 2! * 1!) = 30\r<BR>\n" );
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document.write( "3. **Calculate the probability for the chosen customers:**<BR>\n" );
document.write( "   * The probability that the first two defective components go to the first chosen customer is (1/5) * (1/5).<BR>\n" );
document.write( "   * The probability that the next two defective components go to the second chosen customer is (1/5) * (1/5).<BR>\n" );
document.write( "   * The last defective component goes to the remaining customer with probability 1/5.\r<BR>\n" );
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document.write( "4. **Calculate the probability for the remaining components:**<BR>\n" );
document.write( "   * The remaining 45 components must be distributed so that the two chosen \"no defective\" customers only receive components from the remaining 45.<BR>\n" );
document.write( "   * Probability for remaining components = (45! / (22! * 23!)) * (2/5)^22 * (2/5)^23\r<BR>\n" );
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document.write( "5. **Calculate the overall probability:**<BR>\n" );
document.write( "   * Probability = (Ways to choose customers) * (Ways to assign defective components) * (Probability for chosen customers) * (Probability for remaining components)<BR>\n" );
document.write( "   * Probability = 10 * 3 * 30 * (1/5)^5 * (45! / (22! * 23!)) * (2/5)^22 * (2/5)^23<BR>\n" );
document.write( "   * This calculation is quite complex and best done with a calculator or computer. The approximate result is **0.114**. </SPAN>\n" );
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