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    "# Practice Problems\n",
    "### Lecture 15\n",
    "Answer each number in a separate cell\n",
    "\n",
    "Rename this notebook with your last name and the lecture \n",
    "    \n",
    "    ex. Cych_B_15\n",
    "    \n",
    "Turn-in this notebook on Canvas"
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    "## 1.  Binomial functions\n",
    "- Assume that the probability of getting a winning lottery ticket ($p$) is 1 in 20 and you have money to purchase up to 10 tickets ($n$). \n",
    "    - Calculate the probability of purchasing one winning lottery ticket. \n",
    "    - Create a list of probabilites. The probability of purchasing 1 winning ticket, 2 winning tickets, 3 winning tickets, ... up to 10 winning tickets. \n",
    "    - Plot the probability against the number of winning tickets purchased as a green bar plot. Label both axes\n",
    "\n",
    "\n",
    "## 2. Monte Carlo simulations with stats.binom( )\n",
    "- Again, assume that the probability of getting a winning lottery ticket ($p$) is 1 in 20 and you have money to purchase up to 10 tickets ($n$). \n",
    "    - Run 100 simulations ($Nmc$) of the scenario \n",
    "    - Plot the simulated results as a histogram and the theoretical distribution as a line graph\n",
    "    - Add a title to the plot \n",
    "    - Add a legend\n",
    "    - Add a label to the x-axis and y-axis\n",
    "\n",
    "\n",
    "## 3. Uniform distributions\n",
    "\n",
    "- Calculate the theoretical distribution of getting a particular azimuth between 0 and 360 when measuring, for example, the direction of a strike - assume that each result is equally likely (a uniform distribution between 0 and 360) \n",
    "- Perform a  Monte Carlo simulation with $n=30$ trials. \n",
    "- Plot your theoretical and simulated results as a bar and histogram plot respectively.  \n",
    "- Try this again using the random.seed() function.  \n",
    "\n",
    "## 4. Normal distributions\n",
    "- Calculate the theoretical distribution of grades on an exam with a mean of 50% and a standard deviation of $\\pm$ 20.\n",
    "- Simulate the results of an exam taken by 35 students\n",
    "- Calculate the mean and standard deviation of your simulated results. \n",
    "- Plot theoretical and simulated results as a bar graph and histogram respectively.\n",
    "- Plot a solid line representing the cutoffs for As, Bs, Cs, Ds, and Fs \n",
    "- Add a title, x-label, y-label, and legend\n",
    "\n",
    "# 5. Log-normal distributions\n",
    "- Simulate a grain size distribution that is drawn from a log normal distribution with 1000 grains and a mean and standard deviation of 10 and 0.1 microns respectively. \n",
    "- Plot the distribution as a histogram (density set to True).\n",
    "- Calculate the mean, standard deviation, expectation and variance of the distribution.  "
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