#!/usr/bin/env python
# coding: utf-8

# # Practice Problems
# ### Lecture 15
# Answer each number in a separate cell
# 
# Rename this notebook with your last name and the lecture 
#     
#     ex. Cych_B_15
#     
# Turn-in this notebook on Canvas

# ## 1.  Binomial functions
# - 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 ($n$). 
#     - Calculate the probability of purchasing one winning lottery ticket. 
#     - Create a list of probabilites. The probability of purchasing 1 winning ticket, 2 winning tickets, 3 winning tickets, ... up to 10 winning tickets. 
#     - Plot the probability against the number of winning tickets purchased as a green bar plot. Label both axes
# 
# 
# ## 2. Monte Carlo simulations with stats.binom( )
# - 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 ($n$). 
#     - Run 100 simulations ($Nmc$) of the scenario 
#     - Plot the simulated results as a histogram and the theoretical distribution as a line graph
#     - Add a title to the plot 
#     - Add a legend
#     - Add a label to the x-axis and y-axis
# 
# 
# ## 3. Uniform distributions
# 
# - 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) 
# - Perform a  Monte Carlo simulation with $n=30$ trials. 
# - Plot your theoretical and simulated results as a bar and histogram plot respectively.  
# - Try this again using the random.seed() function.  
# 
# ## 4. Normal distributions
# - Calculate the theoretical distribution of grades on an exam with a mean of 50% and a standard deviation of $\pm$ 20.
# - Simulate the results of an exam taken by 35 students
# - Calculate the mean and standard deviation of your simulated results. 
# - Plot theoretical and simulated results as a bar graph and histogram respectively.
# - Plot a solid line representing the cutoffs for As, Bs, Cs, Ds, and Fs 
# - Add a title, x-label, y-label, and legend
# 
# # 5. Log-normal distributions
# - 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. 
# - Plot the distribution as a histogram (density set to True).
# - Calculate the mean, standard deviation, expectation and variance of the distribution.  

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