# Practice Problems¶

### Lecture 15¶

Answer each number in a separate cell

Rename this notebook with your last name and the lecture

ex. CychB_16



Turn-in this notebook on TritonEd by the end of class

# 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 random.binomial( )¶

• 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 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.