document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1193726>Question 1193726</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A lottery has 3 types of tickets - 3 that don't win anything, 3 that win 1 dollar and 6 that win 2 dollars. We buy three random tickets. Let S be the sum of winnings, and X - the number of tickets winning 2 dollars. Find the average of random values X and S, their dispersion and correlation coefficient. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #848602 by </B><A HREF=/tutors/aboutme.mpl?userid=proyaop><B>proyaop(69)</B></A><img src=\"/images/stars/stars-08.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=proyaop><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=848602\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Determine the Probability Mass Function (PMF) for X (Number of $2 Tickets)**\r<BR>\n" );
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document.write( "* **Total Tickets:** 3 (no win) + 3 (win $1) + 6 (win $2) = 12 tickets<BR>\n" );
document.write( "* **Probability of winning $2:** 6/12 = 1/2 <BR>\n" );
document.write( "* **Probability of not winning $2:** 6/12 = 1/2\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 tickets)<BR>\n" );
document.write( "            * p = probability of success (probability of drawing a $2 ticket = 1/2)<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) * (1/2)^0 * (1/2)^3 = 1/8 <BR>\n" );
document.write( "    * P(X = 1) = (3C1) * (1/2)^1 * (1/2)^2 = 3/8 <BR>\n" );
document.write( "    * P(X = 2) = (3C2) * (1/2)^2 * (1/2)^1 = 3/8 <BR>\n" );
document.write( "    * P(X = 3) = (3C3) * (1/2)^3 * (1/2)^0 = 1/8 \r<BR>\n" );
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document.write( "**2. 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 * (1/8) + 1 * (3/8) + 2 * (3/8) + 3 * (1/8) <BR>\n" );
document.write( "* E(X) = (3 + 6 + 3) / 8 <BR>\n" );
document.write( "* **E(X) = 1.5**\r<BR>\n" );
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document.write( "**3. 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 - 1.5)² * (1/8) + (1 - 1.5)² * (3/8) + (2 - 1.5)² * (3/8) + (3 - 1.5)² * (1/8) <BR>\n" );
document.write( "* Var(X) = 2.25/8 + 0.25 * (3/8) + 0.25 * (3/8) + 2.25/8 <BR>\n" );
document.write( "* Var(X) = 0.75\r<BR>\n" );
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document.write( "* **Standard Deviation (σ_X) = √Var(X) = √0.75 ≈ 0.866**\r<BR>\n" );
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document.write( "**4. Calculate the Average (Expected Value) of S**\r<BR>\n" );
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document.write( "* **Possible Winnings (S):**<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( "**5. Correlation between X and S**\r<BR>\n" );
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document.write( "* There is a clear relationship between X (number of $2 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( "**6. 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 winning tickets (X) and the total winnings (S) in this lottery scenario.<BR>\n" );
document.write( " </SPAN>\n" );
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