document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1195531>Question 1195531</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Three different phone brands are selected, one at a time from an electronic store containing 6 Samsung phones, 6 I-phones and 6 Techno phones. let X denote the number of I-phones selected in 3 random selections from the store.<BR>\n" );
document.write( "a. Express the probability distribution of X in tabular form.<BR>\n" );
document.write( "b. What is the probability that at least one phone selected will be an I-phone?<BR>\n" );
document.write( "c. what is the probability that none of three selected phones will be an I-phone?<BR>\n" );
document.write( "d. what is the expected number of I-phone selected? <BR>\n" );
document.write( "e. find variance of X and expected value of (4x-2) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848427 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=848427\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Certainly, let's analyze the probability distribution of selecting I-phones.\r<BR>\n" );
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document.write( "**a. Probability Distribution of X in Tabular Form**\r<BR>\n" );
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document.write( "* **Possible Values of X:** <BR>\n" );
document.write( "    * X can take on values 0, 1, 2, or 3.\r<BR>\n" );
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document.write( "* **Total Phones:** 6 Samsung + 6 iPhones + 6 Techno = 18 phones\r<BR>\n" );
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document.write( "* **Calculate Probabilities:**\r<BR>\n" );
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document.write( "    * **P(X = 0):** Probability of selecting no iPhones:<BR>\n" );
document.write( "        * (12/18) * (11/17) * (10/16) = 0.2710\r<BR>\n" );
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document.write( "    * **P(X = 1):** Probability of selecting exactly one iPhone:<BR>\n" );
document.write( "        * (6/18) * (12/17) * (11/16) + (12/18) * (6/17) * (11/16) + (12/18) * (11/17) * (6/16) = 0.4662\r<BR>\n" );
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document.write( "    * **P(X = 2):** Probability of selecting exactly two iPhones:<BR>\n" );
document.write( "        * (6/18) * (5/17) * (12/16) + (6/18) * (12/17) * (5/16) + (12/18) * (6/17) * (5/16) = 0.2268\r<BR>\n" );
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document.write( "    * **P(X = 3):** Probability of selecting all three iPhones:<BR>\n" );
document.write( "        * (6/18) * (5/17) * (4/16) = 0.0357\r<BR>\n" );
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document.write( "* **Probability Distribution Table:**\r<BR>\n" );
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document.write( "| X (Number of iPhones) | P(X) |<BR>\n" );
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document.write( "| 0 | 0.2710 |<BR>\n" );
document.write( "| 1 | 0.4662 |<BR>\n" );
document.write( "| 2 | 0.2268 |<BR>\n" );
document.write( "| 3 | 0.0357 |\r<BR>\n" );
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document.write( "**b. Probability of at least one iPhone**\r<BR>\n" );
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document.write( "* P(X ≥ 1) = P(X = 1) + P(X = 2) + P(X = 3) <BR>\n" );
document.write( "* P(X ≥ 1) = 0.4662 + 0.2268 + 0.0357 = 0.7287\r<BR>\n" );
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document.write( "**c. Probability of no iPhones**\r<BR>\n" );
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document.write( "* P(X = 0) = 0.2710 (already calculated in part a)\r<BR>\n" );
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document.write( "**d. Expected Number of iPhones (E[X])**\r<BR>\n" );
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document.write( "* E[X] = Σ [X * P(X)] <BR>\n" );
document.write( "* E[X] = (0 * 0.2710) + (1 * 0.4662) + (2 * 0.2268) + (3 * 0.0357) <BR>\n" );
document.write( "* E[X] = 0 + 0.4662 + 0.4536 + 0.1071 <BR>\n" );
document.write( "* E[X] = 1\r<BR>\n" );
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document.write( "**e. Variance of X (Var[X])**\r<BR>\n" );
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document.write( "* Var[X] = E[X²] - (E[X])² <BR>\n" );
document.write( "* E[X²] = Σ [X² * P(X)] <BR>\n" );
document.write( "    * E[X²] = (0² * 0.2710) + (1² * 0.4662) + (2² * 0.2268) + (3² * 0.0357) <BR>\n" );
document.write( "    * E[X²] = 0 + 0.4662 + 0.9072 + 0.3213 <BR>\n" );
document.write( "    * E[X²] = 1.6947\r<BR>\n" );
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document.write( "* Var[X] = 1.6947 - (1)² = 0.6947\r<BR>\n" );
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document.write( "**Expected Value of (4X - 2)**\r<BR>\n" );
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document.write( "* E[4X - 2] = 4 * E[X] - 2 <BR>\n" );
document.write( "* E[4X - 2] = 4 * 1 - 2 = 2\r<BR>\n" );
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document.write( "**Summary:**\r<BR>\n" );
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document.write( "* Probability Distribution Table (see part a)<BR>\n" );
document.write( "* P(X ≥ 1) = 0.7287<BR>\n" );
document.write( "* P(X = 0) = 0.2710<BR>\n" );
document.write( "* E[X] = 1<BR>\n" );
document.write( "* Var[X] = 0.6947<BR>\n" );
document.write( "* E[4X - 2] = 2\r<BR>\n" );
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document.write( "I hope this comprehensive explanation is helpful!<BR>\n" );
document.write( " </SPAN>\n" );
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