cs70 4a 2.4 solution

gistfile1.txt

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    "### CS70 Worksheet 4A Problem 2.4"
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    "You have $k$ balls and $n$ bins labelled 1,2,...,$n$ where $n \\geq 2$. You drop each ball uniformly at random into the bins.\n",
    "\n",
    "What is the probability that bin 1 is empty and bin $n$ is non empty?"
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    "Solution: \n",
    "\n",
    "Define the  random variable $B_i$ to be 1 when bin $i$ is not empty and 0 when bin $i$ is empty. Using the definition of the law of total probability:\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[B_1] = \\sum_{B_n} \\mathrm{Pr}[B_1|B_n]\\mathrm{Pr}[B_n]\n",
    "$$\n",
    "\n",
    "Now we'll use the definition of conditional probability $\\mathrm{Pr}[B_1|B_n]\\mathrm{Pr}[B_n] = \\mathrm{Pr}[B_1\\cap B_n]$ to rewrite the summand:\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[B_1] = \\sum_{B_n} \\mathrm{Pr}[B_1\\cap B_n]\n",
    "$$\n",
    "\n",
    "The sum can be expanded as:\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[B_1] =  \\mathrm{Pr}[B_1\\cap (B_n=1)] + \\mathrm{Pr}[B_1\\cap (B_n=0)]\n",
    "$$\n",
    "\n",
    "We're interested in an event in which $B_1=0$ so we will substitute that in for $B_1$:\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[B_1=0] =  \\mathrm{Pr}[(B_1=0)\\cap (B_n=1)] + \\mathrm{Pr}[(B_1=0)\\cap (B_n=0)]\n",
    "$$\n",
    "\n",
    "Rearranging the above equation we get:\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[(B_1=0)\\cap (B_n=1)] =\\mathrm{Pr}[B_1=0] - \\mathrm{Pr}[(B_1=0)\\cap (B_n=0)]\n",
    "$$\n",
    "\n",
    "From the previous parts of this problem we know\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[(B_1=0)\\cap (B_n=0)] = \\left(\\frac{n-2}{n}\\right)^k\n",
    "$$\n",
    "\n",
    "and\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[(B_1=0)] = \\left(\\frac{n-1}{n}\\right)^k\n",
    "$$\n",
    "\n",
    "so\n",
    "\n",
    "$$\n",
    "\\mathrm{Pr}[(B_1=0)\\cap (B_n=1)] =\\left(\\frac{n-1}{n}\\right)^k -  \\left(\\frac{n-2}{n}\\right)^k\n",
    "$$"
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