document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1166229>Question 1166229</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Urn A containes 5 red and 3 white marbles and urn B contain 1 red and 2 white marbles a fair die is tossed if a 3 or 6 appears, the marble is drawn from urn B and put into urn A and then a marble is drawn from urn A otherwise a marble is drawn from urn A and put into urn B and a marble is drawn from urn B what is the probability that <BR>\n" );
document.write( "(1)both marbles are red<BR>\n" );
document.write( "(2) both marbles are white </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852862 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=852862\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> This is a probability problem involving two stages of drawing marbles, where the movement of the first marble depends on the outcome of a fair die roll.\r<BR>\n" );
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document.write( "Let's define the events and initial probabilities:\r<BR>\n" );
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document.write( "## 🎲 Initial Probabilities and Urn Contents\r<BR>\n" );
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document.write( "### Urn Contents<BR>\n" );
document.write( "* **Urn A:** $5R, 3W \implies \text{Total } = 8$<BR>\n" );
document.write( "* **Urn B:** $1R, 2W \implies \text{Total } = 3$\r<BR>\n" );
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document.write( "### Die Roll Events<BR>\n" );
document.write( "Let $D$ be the event of rolling a fair six-sided die.<BR>\n" );
document.write( "* **Event $E_B$ (Draw from B first):** A 3 or 6 appears.<BR>\n" );
document.write( "    $$P(E_B) = P(3) + P(6) = \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \mathbf{\frac{1}{3}}$$<BR>\n" );
document.write( "* **Event $E_A$ (Draw from A first):** Not a 3 or 6 (i.e., 1, 2, 4, or 5).<BR>\n" );
document.write( "    $$P(E_A) = 1 - P(E_B) = 1 - \frac{1}{3} = \mathbf{\frac{2}{3}}$$\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 1. Probability that Both Marbles are Red ($R_1$ and $R_2$)\r<BR>\n" );
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document.write( "Let $M_1$ be the first marble drawn (and moved) and $M_2$ be the second marble drawn. $R_1$ and $R_2$ denote drawing a red marble in the first and second draws, respectively.\r<BR>\n" );
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document.write( "We use the Law of Total Probability, splitting the calculation based on the die roll outcome ($E_B$ or $E_A$).<BR>\n" );
document.write( "$$P(R_1 \cap R_2) = P(R_1 \cap R_2 | E_B) \cdot P(E_B) + P(R_1 \cap R_2 | E_A) \cdot P(E_A)$$\r<BR>\n" );
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document.write( "### Case 1: $E_B$ Occurs (Move B $\to$ A, then Draw A)\r<BR>\n" );
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document.write( "1.  **Draw $M_1$ from Urn B (Red):** $P(R_1 | E_B) = \frac{1}{3}$<BR>\n" );
document.write( "2.  **Move $R_1$ to Urn A:** Urn A contents become $6R, 3W$ (Total 9).<BR>\n" );
document.write( "3.  **Draw $M_2$ from Urn A (Red):** $P(R_2 | R_1 \text{ and } E_B) = \frac{6}{9} = \frac{2}{3}$\r<BR>\n" );
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document.write( "$$P(R_1 \cap R_2 | E_B) = P(R_1 | E_B) \cdot P(R_2 | R_1 \text{ and } E_B) = \frac{1}{3} \cdot \frac{2}{3} = \frac{2}{9}$$\r<BR>\n" );
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document.write( "### Case 2: $E_A$ Occurs (Move A $\to$ B, then Draw B)\r<BR>\n" );
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document.write( "1.  **Draw $M_1$ from Urn A (Red):** $P(R_1 | E_A) = \frac{5}{8}$<BR>\n" );
document.write( "2.  **Move $R_1$ to Urn B:** Urn B contents become $2R, 2W$ (Total 4).<BR>\n" );
document.write( "3.  **Draw $M_2$ from Urn B (Red):** $P(R_2 | R_1 \text{ and } E_A) = \frac{2}{4} = \frac{1}{2}$\r<BR>\n" );
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document.write( "$$P(R_1 \cap R_2 | E_A) = P(R_1 | E_A) \cdot P(R_2 | R_1 \text{ and } E_A) = \frac{5}{8} \cdot \frac{1}{2} = \frac{5}{16}$$\r<BR>\n" );
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document.write( "### Combine Cases<BR>\n" );
document.write( "$$P(R_1 \cap R_2) = \left(\frac{2}{9} \cdot \frac{1}{3}\right) + \left(\frac{5}{16} \cdot \frac{2}{3}\right)$$<BR>\n" );
document.write( "$$P(R_1 \cap R_2) = \frac{2}{27} + \frac{10}{48} = \frac{2}{27} + \frac{5}{24}$$\r<BR>\n" );
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document.write( "To add these, use the common denominator $LCM(27, 24) = 216$:<BR>\n" );
document.write( "$$P(R_1 \cap R_2) = \frac{2 \cdot 8}{216} + \frac{5 \cdot 9}{216} = \frac{16}{216} + \frac{45}{216} = \frac{61}{216}$$\r<BR>\n" );
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document.write( "The probability that both marbles are red is **$\frac{61}{216}$**.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Probability that Both Marbles are White ($W_1$ and $W_2$)\r<BR>\n" );
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document.write( "Let $W_1$ and $W_2$ denote drawing a white marble in the first and second draws, respectively.\r<BR>\n" );
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document.write( "$$P(W_1 \cap W_2) = P(W_1 \cap W_2 | E_B) \cdot P(E_B) + P(W_1 \cap W_2 | E_A) \cdot P(E_A)$$\r<BR>\n" );
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document.write( "### Case 1: $E_B$ Occurs (Move B $\to$ A, then Draw A)\r<BR>\n" );
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document.write( "1.  **Draw $M_1$ from Urn B (White):** $P(W_1 | E_B) = \frac{2}{3}$<BR>\n" );
document.write( "2.  **Move $W_1$ to Urn A:** Urn A contents become $5R, 4W$ (Total 9).<BR>\n" );
document.write( "3.  **Draw $M_2$ from Urn A (White):** $P(W_2 | W_1 \text{ and } E_B) = \frac{4}{9}$\r<BR>\n" );
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document.write( "$$P(W_1 \cap W_2 | E_B) = P(W_1 | E_B) \cdot P(W_2 | W_1 \text{ and } E_B) = \frac{2}{3} \cdot \frac{4}{9} = \frac{8}{27}$$\r<BR>\n" );
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document.write( "### Case 2: $E_A$ Occurs (Move A $\to$ B, then Draw B)\r<BR>\n" );
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document.write( "1.  **Draw $M_1$ from Urn A (White):** $P(W_1 | E_A) = \frac{3}{8}$<BR>\n" );
document.write( "2.  **Move $W_1$ to Urn B:** Urn B contents become $1R, 3W$ (Total 4).<BR>\n" );
document.write( "3.  **Draw $M_2$ from Urn B (White):** $P(W_2 | W_1 \text{ and } E_A) = \frac{3}{4}$\r<BR>\n" );
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document.write( "$$P(W_1 \cap W_2 | E_A) = P(W_1 | E_A) \cdot P(W_2 | W_1 \text{ and } E_A) = \frac{3}{8} \cdot \frac{3}{4} = \frac{9}{32}$$\r<BR>\n" );
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document.write( "### Combine Cases<BR>\n" );
document.write( "$$P(W_1 \cap W_2) = \left(\frac{8}{27} \cdot \frac{1}{3}\right) + \left(\frac{9}{32} \cdot \frac{2}{3}\right)$$<BR>\n" );
document.write( "$$P(W_1 \cap W_2) = \frac{8}{81} + \frac{18}{96} = \frac{8}{81} + \frac{3}{16}$$\r<BR>\n" );
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document.write( "To add these, use the common denominator $LCM(81, 16) = 1296$:<BR>\n" );
document.write( "$$P(W_1 \cap W_2) = \frac{8 \cdot 16}{1296} + \frac{3 \cdot 81}{1296} = \frac{128}{1296} + \frac{243}{1296} = \frac{371}{1296}$$\r<BR>\n" );
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document.write( "The probability that both marbles are white is **$\frac{371}{1296}$**. </SPAN>\n" );
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