document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193954>Question 1193954</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In the lottery there are 6 tickets without winning, 4 tickets with 1 euro winning and 6 tickets with 2 euros winning. Person buys three tickets. S is amount won and X is the number of tickets purchased with 2 euros win. Find the averages, dispersions and correlation of the random variables X and S. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848549 by </B><A HREF=/tutors/aboutme.mpl?userid=parmen><B>parmen(42)</B></A><img src=\"/images/stars/stars-07.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=parmen><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=848549\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Define Random Variables**\r<BR>\n" );
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document.write( "* **X:** The number of 2-euro winning tickets among the three purchased. <BR>\n" );
document.write( "    * X can take on values: 0, 1, 2, or 3.\r<BR>\n" );
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document.write( "* **S:** The total amount won from the three tickets.\r<BR>\n" );
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document.write( "**2. Probability Mass Function (PMF) for X**\r<BR>\n" );
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document.write( "* **Total tickets:** 6 (no win) + 4 (1 euro) + 6 (2 euro) = 16 tickets<BR>\n" );
document.write( "* **Probability of drawing a 2-euro ticket:** 6/16 = 3/8 <BR>\n" );
document.write( "* **Probability of drawing a non-2-euro ticket:** 10/16 = 5/8\r<BR>\n" );
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document.write( "* **Use the Binomial Probability Mass Function:** <BR>\n" );
document.write( "    * P(X = k) = (nCk) * p^k * (1-p)^(n-k) <BR>\n" );
document.write( "        * where:<BR>\n" );
document.write( "            * n = number of trials (3 tickets)<BR>\n" );
document.write( "            * k = number of successes (number of 2-euro tickets)<BR>\n" );
document.write( "            * p = probability of success (probability of drawing a 2-euro ticket = 3/8)<BR>\n" );
document.write( "            * nCk = binomial coefficient (number of combinations of n items taken k at a time) \r<BR>\n" );
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document.write( "    * P(X = 0) = (3C0) * (3/8)^0 * (5/8)^3 = 125/512 <BR>\n" );
document.write( "    * P(X = 1) = (3C1) * (3/8)^1 * (5/8)^2 = 225/512 <BR>\n" );
document.write( "    * P(X = 2) = (3C2) * (3/8)^2 * (5/8)^1 = 135/512 <BR>\n" );
document.write( "    * P(X = 3) = (3C3) * (3/8)^3 * (5/8)^0 = 27/512\r<BR>\n" );
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document.write( "**3. Calculate the Average (Expected Value) of X**\r<BR>\n" );
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document.write( "* E(X) = Σ [k * P(X = k)] <BR>\n" );
document.write( "* E(X) = 0 * (125/512) + 1 * (225/512) + 2 * (135/512) + 3 * (27/512) <BR>\n" );
document.write( "* E(X) = (225 + 270 + 81) / 512 <BR>\n" );
document.write( "* **E(X) = 9/8 = 1.125**\r<BR>\n" );
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document.write( "**4. Calculate the Variance of X**\r<BR>\n" );
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document.write( "* Var(X) = Σ [(k - E(X))² * P(X = k)]<BR>\n" );
document.write( "* Var(X) = (0 - 1.125)² * (125/512) + (1 - 1.125)² * (225/512) + (2 - 1.125)² * (135/512) + (3 - 1.125)² * (27/512) <BR>\n" );
document.write( "* Var(X) ≈ 0.4688\r<BR>\n" );
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document.write( "* **Standard Deviation (σ_X) = √Var(X) ≈ 0.685**\r<BR>\n" );
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document.write( "**5. Calculate the Average (Expected Value) of S**\r<BR>\n" );
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document.write( "* **Possible winnings:**<BR>\n" );
document.write( "    * 0 euros (no winning tickets)<BR>\n" );
document.write( "    * 1 euro (one 1-euro ticket, no 2-euro tickets)<BR>\n" );
document.write( "    * 2 euros (two 1-euro tickets, no 2-euro tickets or one 1-euro and one 2-euro ticket)<BR>\n" );
document.write( "    * 3 euros (three 1-euro tickets or one 1-euro and one 2-euro ticket)<BR>\n" );
document.write( "    * 4 euros (two 2-euro tickets and one 1-euro ticket)<BR>\n" );
document.write( "    * 6 euros (three 2-euro tickets)\r<BR>\n" );
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document.write( "* Calculate the probability of each winning scenario (this can be more complex).\r<BR>\n" );
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document.write( "* **E(S) = Σ [s * P(S = s)]** \r<BR>\n" );
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document.write( "* **Note:** Calculating E(S) and its variance would require a more detailed analysis of all possible winning scenarios and their probabilities.\r<BR>\n" );
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document.write( "**6. Correlation between X and S**\r<BR>\n" );
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document.write( "* There is a clear relationship between X (number of 2-euro tickets) and S (total winnings). <BR>\n" );
document.write( "* As X increases, the total winnings (S) generally increase. <BR>\n" );
document.write( "* Therefore, we would expect a **positive correlation** between X and S.\r<BR>\n" );
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document.write( "**7. Calculate the Correlation Coefficient (ρ)**\r<BR>\n" );
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document.write( "* Calculating the correlation coefficient (ρ) between X and S would require:<BR>\n" );
document.write( "    * The joint probability distribution of X and S.<BR>\n" );
document.write( "    * Calculating the covariance between X and S.<BR>\n" );
document.write( "    * Calculating the standard deviation of S (σ_S).\r<BR>\n" );
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document.write( "* **ρ = Cov(X, S) / (σ_X * σ_S)**\r<BR>\n" );
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document.write( "**Key Considerations:**\r<BR>\n" );
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document.write( "* This analysis provides a general framework.<BR>\n" );
document.write( "* Calculating the exact probabilities and expected values for S can be more involved.<BR>\n" );
document.write( "* Using statistical software or programming tools can help with the calculations and simulations.\r<BR>\n" );
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document.write( "This analysis provides a foundation for understanding the relationship between the number of 2-euro winning tickets (X) and the total winnings (S) in this lottery scenario.<BR>\n" );
document.write( " </SPAN>\n" );
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