4312.sagews

#    Here is a quick way to experiment with the "first ace" random variable.  
#    In contrast to the student version, which shows a very clever home-made shuffling algorithm,
#    I was lazy and just searched for an already-done function in Python.  It turns out that the "random" package contains 
#    a command which randomly shuffles a list.
##############################################################



import random     #import the package that contains the shuffle method.

deck=range(13)*4    #define a deck of cards to be the list [0,1,2,...,12] repeated 4 times.  
                    # We will define aces to be the 4 zeros.
                    
#    Now we define the first_ace() random variable.
def first_ace():
    random.shuffle(deck)    # this shuffles deck 'in place;' i.e., the new value of deck is the scrambled one.
    return 1+deck.index(0)    # the index method outputs the first index in the list that has the value 0.
                              # We add 1 to it, because lists start with index 0; but decks of cards start with card #1, etc.

#    Output an unshuffled deck, fresh out of the box.
deck

[0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]

#    Test output of the random variable.
first_ace()

13

#    Test that deck is now scrambled.
deck

[7, 10, 1, 11, 8, 11, 10, 12, 11, 12, 5, 6, 0, 7, 3, 2, 8, 7, 10, 8, 0, 9, 4, 12, 12, 10, 3, 6, 3, 3, 4, 9, 1, 2, 6, 4, 5, 11, 5, 0, 1, 7, 5, 9, 2, 0, 4, 1, 6, 8, 2, 9]

#    Now do a heavy-duty test (100,000 trials).

# Create a list of 100000 outputs of the RV.
outputs = [first_ace() for _ in range(100000)]    

# Compute the average by adding these numbers, and dividing by 100000.
# Note the decimal point; that way the variable 'avg' is automatically a float
# rather than an fraction.  
avg =sum(outputs)/100000.

print(avg)   # Note that it is close to 10.6.

10.6043600000000

#    Finally, we display a the frequencies of the various outputs and plot them.
#    For example, how many times did the RV output the value 2?  This is easily 
#    found using the count() method.

outputs.count(2)

7295

#    Now we will make a list of all the output frequencies and plot them.
#    Note that the RV theoretically outputs values in the range 1, 2,..., 49, inclusive,
#    (And remember that the Python object range(1,50) is precisely the range 1,2,...,49, inclusive.)

freq = [outputs.count(i) for i in range(1,50)]

# This next command plots the values in the list.
# It is clear that empirically, the most frequent outcome is 0, and it is indeed about 1/13 of the 100000 trials.
list_plot(freq)

# If you are very observant, you will notice that x-axis values of the plot started with 0,
# which is because Python lists start with index 0.  We can remedy this in a number of ways, but I will leave them to you!