document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1184501>Question 1184501</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A  teacher wants to select randomly a  student from a  group of  4  students.  The names of the students are Anna, Maria, Alex, Ivan.<BR>\n" );
document.write( "(a) Define the set of basic outcomes for this problem. What is the probability of each outcome? <BR>\n" );
document.write( "(b) Consider the random events A = “the name of the selected student starts with A”, B = “the name of the selected student ends with a”. What outcomes do they consist of? What are their probabilities?<BR>\n" );
document.write( "(c) Find the probabilities of the events A∪B, A∩B, A\B, (notA)∪(not B), (not A)∩(not B).<BR>\n" );
document.write( "Include Venn diagrams(“circles”) for each of 5 cases in(c); shade the areas in question. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849842 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=849842\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's break down this probability problem with Venn diagrams.\r<BR>\n" );
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document.write( "**(a) Basic Outcomes and Probabilities:**\r<BR>\n" );
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document.write( "*   **Set of Basic Outcomes:** {Anna, Maria, Alex, Ivan}<BR>\n" );
document.write( "*   **Probability of each outcome:** Since the selection is random, each student has an equal chance of being chosen. Therefore, the probability of each outcome is 1/4 or 0.25.\r<BR>\n" );
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document.write( "**(b) Events A and B:**\r<BR>\n" );
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document.write( "*   **Event A (\"starts with A\"):** {Anna, Alex}  P(A) = 2/4 = 1/2 = 0.5<BR>\n" );
document.write( "*   **Event B (\"ends with a\"):** {Anna, Maria}  P(B) = 2/4 = 1/2 = 0.5\r<BR>\n" );
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document.write( "**(c) Combined Probabilities and Venn Diagrams:**\r<BR>\n" );
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document.write( "Here's a breakdown of each combined probability along with a description of the corresponding Venn diagram.  Imagine two overlapping circles. One represents event A (starts with A), and the other represents event B (ends with a).\r<BR>\n" );
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document.write( "1.  **A ∪ B (A or B or both):**\r<BR>\n" );
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document.write( "    *   **Outcomes:** {Anna, Maria, Alex}<BR>\n" );
document.write( "    *   **Probability:** P(A ∪ B) = 3/4 = 0.75<BR>\n" );
document.write( "    *   **Venn Diagram:** Shade *both* circles and the overlapping area in the center. This represents all outcomes in A, all outcomes in B, and any outcomes that are in both.\r<BR>\n" );
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document.write( "2.  **A ∩ B (A and B):**\r<BR>\n" );
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document.write( "    *   **Outcomes:** {Anna}<BR>\n" );
document.write( "    *   **Probability:** P(A ∩ B) = 1/4 = 0.25<BR>\n" );
document.write( "    *   **Venn Diagram:** Shade *only* the overlapping area in the center of the two circles. This represents the outcomes that are in *both* A and B.\r<BR>\n" );
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document.write( "3.  **A \ B (A but not B):**\r<BR>\n" );
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document.write( "    *   **Outcomes:** {Alex}<BR>\n" );
document.write( "    *   **Probability:** P(A \ B) = 1/4 = 0.25<BR>\n" );
document.write( "    *   **Venn Diagram:** Shade the part of circle A that *does not* overlap with circle B. This represents outcomes that are in A but *not* in B.\r<BR>\n" );
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document.write( "4.  **(not A) ∪ (not B) (Not A or Not B):**\r<BR>\n" );
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document.write( "    *   **Outcomes:** {Maria, Ivan, Alex}<BR>\n" );
document.write( "    *   **Probability:** P((not A) ∪ (not B)) = 3/4 = 0.75  (This is the same as not(A and B), by De Morgan's Law)<BR>\n" );
document.write( "    *   **Venn Diagram:** Shade everything *outside* the area where the two circles overlap.  This represents outcomes that are *not* in A, outcomes that are *not* in B, and all outcomes that are not in the intersection.\r<BR>\n" );
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document.write( "5.  **(not A) ∩ (not B) (Not A and Not B):**\r<BR>\n" );
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document.write( "    *   **Outcomes:** {Ivan}<BR>\n" );
document.write( "    *   **Probability:** P((not A) ∩ (not B)) = 1/4 = 0.25 (Only Ivan satisfies neither condition.)<BR>\n" );
document.write( "    *   **Venn Diagram:** Shade the area *outside* of both circles. This represents the outcomes that are *neither* in A *nor* in B.<BR>\n" );
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