document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1178959>Question 1178959</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ownership status Size<BR>\n" );
document.write( "          Small Large<BR>\n" );
document.write( "Private    68    45<BR>\n" );
document.write( "Public     33   76\r<BR>\n" );
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document.write( "(a) Suppose a company is selected at random. Then compute the probability that<BR>\n" );
document.write( "I. The company is private or it is large. <BR>\n" );
document.write( "II. The company is small and publicly owned. <BR>\n" );
document.write( "(b) Are being publicly owned and being a large company independent events? <BR>\n" );
document.write( "(c) What is the value of P(small and large? Explain your answer. <BR>\n" );
document.write( "(d) If two small firms at randomly selected what is the probability that both are private? <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 #850273 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</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=850273\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this problem step-by-step:\r<BR>\n" );
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document.write( "**1. Create a Total Table:**\r<BR>\n" );
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document.write( "| Ownership Status | Small | Large | Total |<BR>\n" );
document.write( "|------------------|-------|-------|-------|<BR>\n" );
document.write( "| Private          | 68    | 45    | 113   |<BR>\n" );
document.write( "| Public           | 33    | 76    | 109   |<BR>\n" );
document.write( "| Total            | 101   | 121   | 222   |\r<BR>\n" );
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document.write( "**a) Probability Calculations:**\r<BR>\n" );
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document.write( "* **I. The company is private or it is large.**<BR>\n" );
document.write( "    * P(Private) = 113/222<BR>\n" );
document.write( "    * P(Large) = 121/222<BR>\n" );
document.write( "    * P(Private AND Large) = 45/222<BR>\n" );
document.write( "    * P(Private OR Large) = P(Private) + P(Large) - P(Private AND Large)<BR>\n" );
document.write( "    * P(Private OR Large) = (113/222) + (121/222) - (45/222) = 189/222 = 63/74 ≈ 0.8514\r<BR>\n" );
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document.write( "* **II. The company is small and publicly owned.**<BR>\n" );
document.write( "    * P(Small AND Public) = 33/222 = 11/74 ≈ 0.1486\r<BR>\n" );
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document.write( "**b) Independence of Public Ownership and Large Size:**\r<BR>\n" );
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document.write( "* To check for independence, we need to see if P(Public AND Large) = P(Public) * P(Large).<BR>\n" );
document.write( "* P(Public AND Large) = 76/222 = 38/111 ≈ 0.3423<BR>\n" );
document.write( "* P(Public) = 109/222 ≈ 0.4910<BR>\n" );
document.write( "* P(Large) = 121/222 ≈ 0.5450<BR>\n" );
document.write( "* P(Public) * P(Large) = (109/222) * (121/222) ≈ 0.2677<BR>\n" );
document.write( "* Since P(Public AND Large) ≠ P(Public) * P(Large), the events are **not independent**.\r<BR>\n" );
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document.write( "**c) P(Small and Large)?**\r<BR>\n" );
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document.write( "* P(Small AND Large) = 0/222 = 0<BR>\n" );
document.write( "* This is because a company cannot be both \"small\" and \"large\" simultaneously. These are mutually exclusive categories within the provided data.\r<BR>\n" );
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document.write( "**d) Probability of Two Small Firms Being Private:**\r<BR>\n" );
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document.write( "* P(Small AND Private) = 68/101<BR>\n" );
document.write( "* We need to find the probability of selecting two small firms that are both private.<BR>\n" );
document.write( "* P(1st small firm is private) = 68/101<BR>\n" );
document.write( "* P(2nd small firm is private, given the 1st was private) = 67/100 (since we assume selections are without replacement)<BR>\n" );
document.write( "* P(Both are private) = (68/101) * (67/100) = 4556/10100 ≈ 0.4511\r<BR>\n" );
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document.write( "**Answers:**\r<BR>\n" );
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document.write( "* **(a) I.** 189/222 or approximately 0.8514<BR>\n" );
document.write( "* **(a) II.** 33/222 or approximately 0.1486<BR>\n" );
document.write( "* **(b)** No, they are not independent.<BR>\n" );
document.write( "* **(c)** 0, because a firm can't be both small and large.<BR>\n" );
document.write( "* **(d)** 4556/10100 or approximately 0.4511<BR>\n" );
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