document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1168570>Question 1168570</A>: <I><SPAN STYLE=\"word-break: break-all;\">  If we roll a die two times, determine whether the following pairs of events are independent or dependent.<BR>\n" );
document.write( "Event A is rolling a 3 on the first die. Event B is getting a sum of more than 6 with the two dice? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #851517 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=851517\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's determine if events A and B are independent or dependent.\r<BR>\n" );
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document.write( "**1. Define the Events**\r<BR>\n" );
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document.write( "* **Event A:** Rolling a 3 on the first die.<BR>\n" );
document.write( "* **Event B:** Getting a sum of more than 6 with the two dice.\r<BR>\n" );
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document.write( "**2. Calculate Probabilities**\r<BR>\n" );
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document.write( "* **P(A):** The probability of rolling a 3 on the first die is 1/6 (since there's one favorable outcome out of six).<BR>\n" );
document.write( "* **P(B):** To find P(B), we need to count the pairs of dice rolls that sum to more than 6.<BR>\n" );
document.write( "    * Possible pairs: (1,6), (2,5), (2,6), (3,4), (3,5), (3,6), (4,3), (4,4), (4,5), (4,6), (5,2), (5,3), (5,4), (5,5), (5,6), (6,1), (6,2), (6,3), (6,4), (6,5), (6,6).<BR>\n" );
document.write( "    * There are 21 pairs that sum to more than 6.<BR>\n" );
document.write( "    * Total possible pairs: 6 * 6 = 36.<BR>\n" );
document.write( "    * P(B) = 21/36 = 7/12.<BR>\n" );
document.write( "* **P(A and B):** The probability of both events A and B occurring.<BR>\n" );
document.write( "    * We need to count the pairs where the first die is a 3 and the sum is greater than 6.<BR>\n" );
document.write( "    * These pairs are: (3,4), (3,5), (3,6).<BR>\n" );
document.write( "    * P(A and B) = 3/36 = 1/12.\r<BR>\n" );
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document.write( "**3. Check for Independence**\r<BR>\n" );
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document.write( "* Two events are independent if P(A and B) = P(A) * P(B).<BR>\n" );
document.write( "* Let's check:<BR>\n" );
document.write( "    * P(A) * P(B) = (1/6) * (7/12) = 7/72<BR>\n" );
document.write( "    * P(A and B) = 1/12 = 6/72\r<BR>\n" );
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document.write( "* Since 7/72 ≠ 6/72, P(A and B) ≠ P(A) * P(B).\r<BR>\n" );
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document.write( "**4. Conclusion**\r<BR>\n" );
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document.write( "* Therefore, events A and B are **dependent**.<BR>\n" );
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