A shipment of 7 television sets contains 2 defective sets. A hotel makes a random purchase of 3 of the sets. Considering that y is the number of non-defective sets purchased by the hotel
(a) What are the values for the set of y?
(b) What is the probability of purchasing 2 non-defective television sets? (Fraction form)
(c) make a probability distribution table.
(a) Since a shipment of 7 television sets contains 2 defective sets and a hotel makes a random purchase of 3 of the sets, for the number "y" of non-defective sets purchased by the hotel we have "y\\in\\{1,2,3\\}".
(b) The probability "p" of purchasing 2 non-defective television sets is
"p=\\frac{{5 \\choose 2}{2\\choose 1}}{7 \\choose 3}=\\frac{\\frac{5!}{2!\\cdot 3!}\\cdot 2}{\\frac{7!}{3!\\cdot 4!}}=\\frac{5!\\cdot 4!\\cdot 2}{2!\\cdot 7!}=\\frac{2\\cdot 3\\cdot 4}{6\\cdot 7}=\\frac{4}{7}."
(c) Let us make a probability distribution table for "y\\in\\{1,2,3\\}".
We already have that "P(y=2)=\\frac{4}{7}".
"P(y=1)=\\frac{{5 \\choose 1}{2\\choose 2}}{7 \\choose 3}=\\frac{5}{\\frac{7!}{3!\\cdot 4!}}=\\frac{5\\cdot 3!\\cdot 4!}{ 7!}=\\frac{5\\cdot 2\\cdot 3}{7\\cdot 6\\cdot 5}=\\frac{1}{7}"
"P(y=3)=\\frac{{5 \\choose 3}}{7 \\choose 3}=\\frac{\\frac{5!}{3!\\cdot 2!}}{\\frac{7!}{3!\\cdot 4!}}=\\frac{5!\\cdot 4!}{2!\\cdot 7!}=\\frac{ 2\\cdot 3\\cdot 4}{2\\cdot 7\\cdot 6}=\\frac{2}{7}"
Therefore, a probability distribution table is the following:
"\\begin{array}{|c|c|c|c|}\n\\hline\n y & 1 & 2 & 3\\\\\n\n p & \\frac{1}{7} & \\frac{4}{7} & \\frac{2}{7} \\\\\n\\hline\n\\end{array}"
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