document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1192443>Question 1192443</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A balanced die is tossed twice. If A=Event (an even number comes up on the first toss), B = Event( an even number comes up on the 2nd toss, and C = Event ( both tosses result in the same number), are the events A B and C<BR>\n" );
document.write( "(a) pairwise Independent?<BR>\n" );
document.write( "(b) independent? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848983 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=848983\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Events**\r<BR>\n" );
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document.write( "* **A:** Event of an even number on the first toss (A = {2, 4, 6})<BR>\n" );
document.write( "* **B:** Event of an even number on the second toss (B = {2, 4, 6})<BR>\n" );
document.write( "* **C:** Event of both tosses resulting in the same number (C = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)})\r<BR>\n" );
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document.write( "**2. Calculate Probabilities**\r<BR>\n" );
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document.write( "* **P(A):** Probability of an even number on the first toss = 3/6 = 1/2<BR>\n" );
document.write( "* **P(B):** Probability of an even number on the second toss = 3/6 = 1/2<BR>\n" );
document.write( "* **P(C):** Probability of both tosses being the same = 6/36 = 1/6<BR>\n" );
document.write( "* **P(A ∩ B):** Probability of even numbers on both tosses = 3/6 * 3/6 = 1/4<BR>\n" );
document.write( "* **P(A ∩ C):** Probability of even number on first toss and both tosses being the same = 3/36 = 1/12<BR>\n" );
document.write( "* **P(B ∩ C):** Probability of even number on second toss and both tosses being the same = 3/36 = 1/12\r<BR>\n" );
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document.write( "**3. Check for Pairwise Independence**\r<BR>\n" );
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document.write( "* **A and B:**<BR>\n" );
document.write( "    * P(A ∩ B) = 1/4 <BR>\n" );
document.write( "    * P(A) * P(B) = (1/2) * (1/2) = 1/4 <BR>\n" );
document.write( "    * Since P(A ∩ B) = P(A) * P(B), events A and B are independent.\r<BR>\n" );
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document.write( "* **A and C:**<BR>\n" );
document.write( "    * P(A ∩ C) = 1/12 <BR>\n" );
document.write( "    * P(A) * P(C) = (1/2) * (1/6) = 1/12 <BR>\n" );
document.write( "    * Since P(A ∩ C) = P(A) * P(C), events A and C are independent.\r<BR>\n" );
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document.write( "* **B and C:**<BR>\n" );
document.write( "    * P(B ∩ C) = 1/12 <BR>\n" );
document.write( "    * P(B) * P(C) = (1/2) * (1/6) = 1/12 <BR>\n" );
document.write( "    * Since P(B ∩ C) = P(B) * P(C), events B and C are independent.\r<BR>\n" );
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document.write( "**4. Check for Mutual Independence**\r<BR>\n" );
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document.write( "* For events to be mutually independent, the following condition must hold:<BR>\n" );
document.write( "    * P(A ∩ B ∩ C) = P(A) * P(B) * P(C)\r<BR>\n" );
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document.write( "* P(A ∩ B ∩ C) = Probability of even numbers on both tosses and both tosses being the same = 3/36 = 1/12<BR>\n" );
document.write( "* P(A) * P(B) * P(C) = (1/2) * (1/2) * (1/6) = 1/24\r<BR>\n" );
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document.write( "* Since P(A ∩ B ∩ C) ≠ P(A) * P(B) * P(C), the events A, B, and C are **not mutually independent**.\r<BR>\n" );
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document.write( "**Conclusion**\r<BR>\n" );
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document.write( "* **(a) Pairwise Independent:** Yes, events A, B, and C are pairwise independent.<BR>\n" );
document.write( "* **(b) Independent:** No, events A, B, and C are not mutually independent.<BR>\n" );
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