document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186890>Question 1186890</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let A, B and C be three arbitrary events. Find expressions for the following events that <BR>\n" );
document.write( "of A, B and C.<BR>\n" );
document.write( "a- Only A occurs, b- All three events occur, c- Non occurs,<BR>\n" );
document.write( "d- At least one occurs, e- Not more than 2 occur, g-One and only one occurs. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849337 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=849337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here are the expressions for the given events using set notation:\r<BR>\n" );
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document.write( "**a. Only A occurs:**\r<BR>\n" );
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document.write( "A ∩ Bᶜ ∩ Cᶜ  (A intersects with the complements of B and C)\r<BR>\n" );
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document.write( "**b. All three events occur:**\r<BR>\n" );
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document.write( "A ∩ B ∩ C  (The intersection of all three events)\r<BR>\n" );
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document.write( "**c. None occurs:**\r<BR>\n" );
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document.write( "Aᶜ ∩ Bᶜ ∩ Cᶜ  (The intersection of the complements of all three events). This can also be written as (A ∪ B ∪ C)ᶜ (De Morgan's Law).\r<BR>\n" );
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document.write( "**d. At least one occurs:**\r<BR>\n" );
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document.write( "A ∪ B ∪ C  (The union of all three events)\r<BR>\n" );
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document.write( "**e. Not more than 2 occur:**\r<BR>\n" );
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document.write( "This is equivalent to \"none occur\" or \"exactly one occurs\" or \"exactly two occur\". It is the complement of \"all three occur\".\r<BR>\n" );
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document.write( "(A ∩ B ∩ C)ᶜ\r<BR>\n" );
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document.write( "Alternatively, we can express it as the union of the cases where no events occur, only one occurs, and exactly two occur.\r<BR>\n" );
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document.write( "(Aᶜ ∩ Bᶜ ∩ Cᶜ) ∪ (A ∩ Bᶜ ∩ Cᶜ) ∪ (Aᶜ ∩ B ∩ Cᶜ) ∪ (Aᶜ ∩ Bᶜ ∩ C) ∪ (A ∩ B ∩ Cᶜ) ∪ (A ∩ Bᶜ ∩ C) ∪ (Aᶜ ∩ B ∩ C)\r<BR>\n" );
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document.write( "**f. One and only one occurs:**\r<BR>\n" );
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document.write( "(A ∩ Bᶜ ∩ Cᶜ) ∪ (Aᶜ ∩ B ∩ Cᶜ) ∪ (Aᶜ ∩ Bᶜ ∩ C) (The union of the three mutually exclusive scenarios where only one of the events occurs).<BR>\n" );
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