document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1190338>Question 1190338</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Ten thousand lottery tickets are sold at $10 each. First prize is $10000. There are 3 second prizes at $5000 each. There are 10 third prizes at $1000 each. If X is the discrete random variable that measures the gain from buying a lottery ticket, then<BR>\n" );
document.write( " a.	What is the probability distribution of X? <BR>\n" );
document.write( "b.	What is the expected value of X? <BR>\n" );
document.write( "c.	What is the standard deviation of X? \r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849255 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=849255\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to break down this lottery ticket problem:\r<BR>\n" );
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document.write( "**a. Probability Distribution of X (Gain):**\r<BR>\n" );
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document.write( "First, we need to define the possible values of X (the gain).  Remember, gain is what you win *minus* the cost of the ticket ($10).\r<BR>\n" );
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document.write( "*   **First Prize:** Gain = $10,000 - $10 = $9,990<BR>\n" );
document.write( "*   **Second Prize:** Gain = $5,000 - $10 = $4,990<BR>\n" );
document.write( "*   **Third Prize:** Gain = $1,000 - $10 = $990<BR>\n" );
document.write( "*   **No Prize:** Gain = $0 - $10 = -$10\r<BR>\n" );
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document.write( "Now, let's calculate the probabilities of each outcome:\r<BR>\n" );
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document.write( "*   **P(X = $9,990):** 1 ticket out of 10,000 wins the first prize, so the probability is 1/10,000 = 0.0001<BR>\n" );
document.write( "*   **P(X = $4,990):** 3 tickets out of 10,000 win a second prize, so the probability is 3/10,000 = 0.0003<BR>\n" );
document.write( "*   **P(X = $990):** 10 tickets out of 10,000 win a third prize, so the probability is 10/10,000 = 0.001<BR>\n" );
document.write( "*   **P(X = -$10):** The remaining tickets (10,000 - 1 - 3 - 10 = 9,986) win no prize, so the probability is 9,986/10,000 = 0.9986\r<BR>\n" );
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document.write( "Here's the probability distribution:\r<BR>\n" );
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document.write( "| Gain (X) | Probability P(X) |<BR>\n" );
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document.write( "| $9,990 | 0.0001 |<BR>\n" );
document.write( "| $4,990 | 0.0003 |<BR>\n" );
document.write( "| $990 | 0.001 |<BR>\n" );
document.write( "| -$10 | 0.9986 |\r<BR>\n" );
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document.write( "**b. Expected Value of X:**\r<BR>\n" );
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document.write( "The expected value (E[X]) is calculated as:\r<BR>\n" );
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document.write( "E[X] = Σ [x * P(x)]  (summed over all possible values of x)\r<BR>\n" );
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document.write( "E[X] = ($9,990 * 0.0001) + ($4,990 * 0.0003) + ($990 * 0.001) + (-$10 * 0.9986)<BR>\n" );
document.write( "E[X] = $0.999 + $1.497 + $0.99 - $9.986<BR>\n" );
document.write( "E[X] = -$6.50\r<BR>\n" );
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document.write( "The expected value of X is -$6.50. This means that on average, you can expect to lose $6.50 for each lottery ticket you buy.\r<BR>\n" );
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document.write( "**c. Standard Deviation of X:**\r<BR>\n" );
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document.write( "1.  **Calculate E[X²]:**<BR>\n" );
document.write( "    E[X²] = Σ [x² * P(x)]<BR>\n" );
document.write( "    E[X²] = (9990² * 0.0001) + (4990² * 0.0003) + (990² * 0.001) + (-10² * 0.9986)<BR>\n" );
document.write( "    E[X²] = 99800.1 + 7485.003 + 980.1 + 99.86<BR>\n" );
document.write( "    E[X²] ≈ 108365.06\r<BR>\n" );
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document.write( "2.  **Calculate Variance (Var[X]):**<BR>\n" );
document.write( "    Var[X] = E[X²] - (E[X])²<BR>\n" );
document.write( "    Var[X] = 108365.06 - (-6.50)²<BR>\n" );
document.write( "    Var[X] = 108365.06 - 42.25<BR>\n" );
document.write( "    Var[X] ≈ 108322.81\r<BR>\n" );
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document.write( "3.  **Calculate Standard Deviation (SD[X]):**<BR>\n" );
document.write( "    SD[X] = √Var[X]<BR>\n" );
document.write( "    SD[X] = √108322.81<BR>\n" );
document.write( "    SD[X] ≈ $329.12\r<BR>\n" );
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document.write( "The standard deviation of X is approximately $329.12.<BR>\n" );
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