document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1163935>Question 1163935</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A box contains 6 balls, 3 are blue and 3 are red. Balls are drawn successively without replacement. If random variable X is number of draws until last red ball is obtained. Find the probability density function of random variable X? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #854366 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2264)</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=854366\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import math\r<BR>\n" );
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document.write( "def nCr(n, r):<BR>\n" );
document.write( "    if r < 0 or r > n:<BR>\n" );
document.write( "        return 0<BR>\n" );
document.write( "    return math.comb(n, r)\r<BR>\n" );
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document.write( "def calculate_pmf():<BR>\n" );
document.write( "    N = 6<BR>\n" );
document.write( "    R = 3<BR>\n" );
document.write( "    results = {}<BR>\n" );
document.write( "    total_combinations = nCr(N, R)<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    for k in range(3, 7):<BR>\n" );
document.write( "        # Ways to have 2 red balls in first k-1 slots * 1 red ball in kth slot<BR>\n" );
document.write( "        ways = nCr(k-1, 2)<BR>\n" );
document.write( "        prob = ways / total_combinations<BR>\n" );
document.write( "        results[k] = (ways, prob)<BR>\n" );
document.write( "    <BR>\n" );
document.write( "    return results, total_combinations\r<BR>\n" );
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document.write( "pmf, total = calculate_pmf()<BR>\n" );
document.write( "print(f\"{total=}\")<BR>\n" );
document.write( "for k, (ways, prob) in pmf.items():<BR>\n" );
document.write( "    print(f\"X={k}: Ways={ways}, Prob={prob}\")\r<BR>\n" );
document.write( "<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "total=20<BR>\n" );
document.write( "X=3: Ways=1, Prob=0.05<BR>\n" );
document.write( "X=4: Ways=3, Prob=0.15<BR>\n" );
document.write( "X=5: Ways=6, Prob=0.3<BR>\n" );
document.write( "X=6: Ways=10, Prob=0.5\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "To find the probability mass function (PMF) of the random variable $X$, which represents the number of draws until the last red ball is obtained, we can use combinatorial counting.\r<BR>\n" );
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document.write( "### 1. Identify the Parameters<BR>\n" );
document.write( "* Total number of balls: $N = 6$<BR>\n" );
document.write( "* Number of red balls: $R = 3$<BR>\n" );
document.write( "* Number of blue balls: $B = 3$<BR>\n" );
document.write( "* Random variable $X$: The draw number at which the **3rd** red ball is pulled.\r<BR>\n" );
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document.write( "The range of $X$ is $\{3, 4, 5, 6\}$.<BR>\n" );
document.write( "* $X=3$ is the earliest possible draw (if the first three balls are all red).<BR>\n" );
document.write( "* $X=6$ is the latest possible draw (if two blue balls are drawn before the final red ball).\r<BR>\n" );
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document.write( "### 2. Logic for $P(X=k)$<BR>\n" );
document.write( "For the $k$-th draw to contain the last red ball, two conditions must be met:<BR>\n" );
document.write( "1.  The $k$-th ball drawn must be **Red**.<BR>\n" );
document.write( "2.  In the previous $k-1$ draws, there must have been exactly **2 Red balls** (and $k-3$ Blue balls).\r<BR>\n" );
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document.write( "The total number of ways to arrange 3 red balls and 3 blue balls in 6 positions is:<BR>\n" );
document.write( "$$\text{Total Outcomes} = \binom{6}{3} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20$$\r<BR>\n" );
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document.write( "The number of successful outcomes for a specific $k$ is the number of ways to choose 2 positions for the red balls out of the first $k-1$ positions:<BR>\n" );
document.write( "$$\text{Favorable Outcomes} = \binom{k-1}{2}$$\r<BR>\n" );
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document.write( "### 3. Calculate the Probabilities<BR>\n" );
document.write( "We apply the formula $P(X=k) = \frac{\binom{k-1}{2}}{20}$ for each $k$:\r<BR>\n" );
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document.write( "* **For $X=3$:**<BR>\n" );
document.write( "    $$P(X=3) = \frac{\binom{3-1}{2}}{20} = \frac{\binom{2}{2}}{20} = \frac{1}{20} = 0.05$$<BR>\n" );
document.write( "* **For $X=4$:**<BR>\n" );
document.write( "    $$P(X=4) = \frac{\binom{4-1}{2}}{20} = \frac{\binom{3}{2}}{20} = \frac{3}{20} = 0.15$$<BR>\n" );
document.write( "* **For $X=5$:**<BR>\n" );
document.write( "    $$P(X=5) = \frac{\binom{5-1}{2}}{20} = \frac{\binom{4}{2}}{20} = \frac{6}{20} = 0.30$$<BR>\n" );
document.write( "* **For $X=6$:**<BR>\n" );
document.write( "    $$P(X=6) = \frac{\binom{6-1}{2}}{20} = \frac{\binom{5}{2}}{20} = \frac{10}{20} = 0.50$$\r<BR>\n" );
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document.write( "### 4. Summary of the Probability Density Function (PMF)\r<BR>\n" );
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document.write( "The probability density (mass) function is given by:\r<BR>\n" );
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document.write( "| $x$ | $P(X=x)$ |<BR>\n" );
document.write( "| :--- | :--- |<BR>\n" );
document.write( "| 3 | $1/20$ ($0.05$) |<BR>\n" );
document.write( "| 4 | $3/20$ ($0.15$) |<BR>\n" );
document.write( "| 5 | $6/20$ ($0.30$) |<BR>\n" );
document.write( "| 6 | $10/20$ ($0.50$) |<BR>\n" );
document.write( "| Other | $0$ |\r<BR>\n" );
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document.write( "**Check:** $0.05 + 0.15 + 0.30 + 0.50 = 1.0$. The sum of probabilities equals 1, confirming the distribution is valid. </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );