document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165742>Question 1165742</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In a lottery game, a player picks 8 numbers from 1 to 48. If 6 of those 8 numbers match those drawn, the player wins third prize. Let's walk through the steps to determine the probability of winning third prize.\r<BR>\n" );
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document.write( "In how many ways can 6 winning numbers be chosen from the possible 8 numbers? \r<BR>\n" );
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document.write( "In how many ways can 2 non-winning numbers be chosen from the pool of all non-winning numbers? \r<BR>\n" );
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document.write( "The number of favorable outcomes would be to multiply the above two answers together, since we want 6 winning numbers and 2 non-winning numbers. What is the number of favorable outcomes? \r<BR>\n" );
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document.write( "In how many ways can you pick any 8 numbers from the pool of 48 numbers? This is your total outcomes. <BR>\n" );
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document.write( "What is the probability of winning third prize? <BR>\n" );
document.write( " (Round to 5 decimal places.) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852937 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2189)</B></A><img src=\"/images/stars/stars-13.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=852937\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a classic probability problem that uses combinations. Let's break down the steps to find the probability of winning the third prize.\r<BR>\n" );
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document.write( "The game parameters are:<BR>\n" );
document.write( "* Total numbers available: **48**<BR>\n" );
document.write( "* Numbers drawn (Winning numbers): **8**<BR>\n" );
document.write( "* Numbers a player picks: **8**<BR>\n" );
document.write( "* Winning Condition (Third Prize): Matching exactly **6** of the 8 drawn numbers.\r<BR>\n" );
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document.write( "This means a favorable outcome requires matching 6 winning numbers and 2 non-winning numbers.\r<BR>\n" );
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document.write( "## 1. Calculating Favorable Outcomes\r<BR>\n" );
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document.write( "First, we need to determine the number of ways to pick 6 winning numbers and 2 non-winning numbers.\r<BR>\n" );
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document.write( "### A. Ways to Choose 6 Winning Numbers\r<BR>\n" );
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document.write( "You need to choose **6** numbers from the **8** winning numbers drawn.<BR>\n" );
document.write( "$$\binom{8}{6} = \frac{8!}{6!(8-6)!} = \frac{8 \times 7}{2 \times 1} = \mathbf{28}$$\r<BR>\n" );
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document.write( "* **In how many ways can 6 winning numbers be chosen from the possible 8 numbers?** $\mathbf{28}$ ways.\r<BR>\n" );
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document.write( "### B. Ways to Choose 2 Non-Winning Numbers\r<BR>\n" );
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document.write( "The total pool is 48 numbers. Since 8 are winning numbers, the number of non-winning numbers is $48 - 8 = 40$. You need to choose **2** numbers from these **40** non-winning numbers.<BR>\n" );
document.write( "$$\binom{40}{2} = \frac{40!}{2!(40-2)!} = \frac{40 \times 39}{2 \times 1} = \mathbf{780}$$\r<BR>\n" );
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document.write( "* **In how many ways can 2 non-winning numbers be chosen from the pool of all non-winning numbers?** $\mathbf{780}$ ways.\r<BR>\n" );
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document.write( "### C. Total Favorable Outcomes\r<BR>\n" );
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document.write( "The total number of favorable outcomes is the product of the ways to choose the winning and non-winning numbers.<BR>\n" );
document.write( "$$\text{Favorable Outcomes} = \binom{8}{6} \times \binom{40}{2} = 28 \times 780 = \mathbf{21,840}$$\r<BR>\n" );
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document.write( "* **The number of favorable outcomes is $\mathbf{21,840}$.**\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Calculating Total Possible Outcomes\r<BR>\n" );
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document.write( "You pick **8** numbers from the pool of **48** numbers. This represents the total number of possible combinations the player can choose.<BR>\n" );
document.write( "$$\text{Total Outcomes} = \binom{48}{8}$$<BR>\n" );
document.write( "$$\binom{48}{8} = \frac{48!}{8!(48-8)!} = \frac{48 \times 47 \times 46 \times 45 \times 44 \times 43 \times 42 \times 41}{8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1}$$<BR>\n" );
document.write( "$$\text{Total Outcomes} = \mathbf{377,370,144}$$\r<BR>\n" );
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document.write( "* **In how many ways can you pick any 8 numbers from the pool of 48 numbers?** $\mathbf{377,370,144}$ ways.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 3. Probability of Winning Third Prize\r<BR>\n" );
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document.write( "The probability is the ratio of favorable outcomes to total possible outcomes.<BR>\n" );
document.write( "$$\text{Probability} = \frac{\text{Favorable Outcomes}}{\text{Total Outcomes}} = \frac{21,840}{377,370,144}$$<BR>\n" );
document.write( "$$\text{Probability} \approx 0.000057874...$$\r<BR>\n" );
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document.write( "Rounding to 5 decimal places: $\mathbf{0.00006}$\r<BR>\n" );
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document.write( "* **What is the probability of winning third prize? (Round to 5 decimal places.)** $\mathbf{0.00006}$ </SPAN>\n" );
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