Probability theory is all around. A common example is in the game of poker or related card games, where players try to calculate the probability of winning a round given the cards they have in their hands. Also, in a business context, probabilities play an important role. Operating in a volatile economy, companies need to take uncertainty into account and this is exactly where probability theory plays a role.
As mentioned in the lesson before, a good understanding of probability starts with understanding of sets and set operations. That's exactly what you'll learn in this lab!
You will be able to:
- Use Python to perform set operations
- Use Python to demonstrate the inclusion/exclusion principle
Let's start with a pretty conceptual example. Let's consider the following sets:
-
$\Omega$ = positive integers between [1, 12] -
$A$ = even numbers between [1, 10] $B = {3,8,11,12}$ $C = {2,3,6,8,9,11}$
a. Illustrate all the sets in a Venn Diagram like the one below. The rectangular shape represents the universal set.
- $ A \cap B$
- $ A \cup C$
$A^c$ - The absolute complement of B
$(A \cup B)^c$ $B \cap C'$ $A\backslash B$ $C \backslash (B \backslash A)$ $(C \cap A) \cup (C \backslash B)$
c. For the remainder of this exercise, let's create sets A, B and C and universal set U in Python and test out the results you came up with. Sets are easy to create in Python. For a guide to the syntax, follow some of the documentation here
# Create set A
A = None
'Type A: {}, A: {}'.format(type(A), A) # "Type A: <class 'set'>, A: {2, 4, 6, 8, 10}"# Create set B
B = None
'Type B: {}, B: {}'.format(type(B), B) # "Type B: <class 'set'>, B: {8, 11, 3, 12}"# Create set C
C = None
'Type C: {}, C: {}'.format(type(C), C) # "Type C: <class 'set'>, C: {2, 3, 6, 8, 9, 11}"# Create universal set U
U = None
'Type U: {}, U: {}'.format(type(U), U) # "Type U: <class 'set'>, U: {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}"Now, verify your answers in section 1 by using the correct methods in Python. To provide a little bit of help, you can find a table with common operations on sets below.
| Method | Equivalent | Result |
|---|---|---|
| s.issubset(t) | s <= t | test whether every element in s is in t |
| s.issuperset(t) | s >= t | test whether every element in t is in s |
| s.union(t) | s |
new set with elements from both s and t |
| s.intersection(t) | s & t | new set with elements common to s and t |
| s.difference(t) | s - t | new set with elements in s but not in t |
| s.symmetric_difference(t) | s ^ t | new set with elements in either s or t but not both |
A_inters_B = None
A_inters_B # {8}A_union_C = None
A_union_C # {2, 3, 4, 6, 8, 9, 10, 11}A_comp = None
A_comp # {1, 3, 5, 7, 9, 11, 12}A_union_B_comp = None
A_union_B_comp # {1, 5, 7, 9}B_inters_C_comp = None
B_inters_C_comp # {12}compl_of_B = None
compl_of_B # {2, 4, 6, 10}C_compl_B_compl_A = None
C_compl_B_compl_A # {2, 6, 8, 9}C_inters_A_union_C_min_B= None
C_inters_A_union_C_min_B # {2, 6, 8, 9}Use A, B and C from exercise one to verify the inclusion exclusion principle in Python. You can use the sets A, B and C as used in the previous exercise.
Recall from the previous lesson that:
Combining these main commands:
| Method | Equivalent | Result |
|---|---|---|
| a.union(b) | A |
new set with elements from both a and b |
| a.intersection(b) | A & B | new set with elements common to a and b |
along with the len(x) function to get to the cardinality of a given x ("|x|").
What you'll do is translate the left hand side of the equation for the inclusion principle in the object left_hand_eq, and the right hand side in the object right_hand_eq and see if the results are the same.
left_hand_eq = None
print(left_hand_eq) # 9 elements in the setright_hand_eq = None
print(right_hand_eq) # 9 elements in the setNone # Use a comparison operator to compare `left_hand_eq` and `right_hand_eq`. Needs to say "True".Mary is preparing for a road trip from her hometown, Boston, to Chicago. She has quite a few pets, yet luckily, so do her friends. They try to make sure that they take care of each other's pets while someone is away on a trip. A month ago, each respective person's pet collection was given by the following three sets:
Nina = set(["Cat","Dog","Rabbit","Donkey","Parrot", "Goldfish"])
Mary = set(["Dog","Chinchilla","Horse", "Chicken"])
Eve = set(["Rabbit", "Turtle", "Goldfish"])In this exercise, you'll be able to use the following operations:
| Operation | Equivalent | Result |
|---|---|---|
| s.update(t) | return set s with elements added from t | |
| s.intersection_update(t) | s &= t | return set s keeping only elements also found in t |
| s.difference_update(t) | s -= t | return set s after removing elements found in t |
| s.symmetric_difference_update(t) | s ^= t | return set s with elements from s or t but not both |
| s.add(x) | add element x to set s | |
| s.remove(x) | remove x from set s | |
| s.discard(x) | removes x from set s if present | |
| s.pop() | remove and return an arbitrary element from s | |
| s.clear() | remove all elements from set s |
Sadly, Eve's turtle passed away last week. Let's update her pet list accordingly.
None
Eve # should be {'Rabbit', 'Goldfish'}This time around, Nina promised to take care of Mary's pets while she's away. But she also wants to make sure her pets are well taken care of. As Nina is already spending a considerable amount of time taking care of her own pets, adding a few more won't make that much of a difference. Nina does want to update her list while Marie is away.
None
Nina # {'Chicken', 'Horse', 'Chinchilla', 'Parrot', 'Rabbit', 'Donkey', 'Dog', 'Cat', 'Goldfish'}Mary, on the other hand, wants to clear her list altogether while away:
None
Mary # set()Look at how many species Nina is taking care of right now.
None
n_species_Nina # 9Taking care of this many pets is weighing heavily on Nina. She remembered Eve had a smaller collection of pets lately, and that's why she asks Eve to take care of the common species. This way, the extra pets are not a huge effort on Eve's behalf. Let's update Nina's pet collection.
None
Nina # 7Taking care of 7 species is something Nina feels comfortable doing!
Mary dropped off her Pet's at Nina's house and finally made her way to the highway. Awesome, her vacation has begun! She's approaching an exit. At the end of this particular highway exit, cars can either turn left (L), go straight (S) or turn right (R). It's pretty busy and there are two cars driving close to her. What you'll do now is create several sets. You won't be using Python here, it's sufficient to write the sets down on paper. A good notion of sets and subsets will help you calculate probabilities in the next lab!
Note: each set of action is what all three cars are doing at any given time
a. Create a set
b. Create a set
c. Create a set
d. Create a set
e. Write down the interpretation and give all possible outcomes for the sets denoted by:
- I.
$D'$ - II.
$C \cap D$ , - III.
$C \cup D$ .
Use set operations to determine which European countries are not in the European Union. You just might have to clean the data first with pandas.
import pandas as pd
#Load Europe and EU
europe = pd.read_excel('Europe_and_EU.xlsx', sheet_name = 'Europe')
eu = pd.read_excel('Europe_and_EU.xlsx', sheet_name = 'EU')
#Use pandas to remove any whitespace from nameseurope.head(3) #preview dataframe
eu.head(3)# Your code comes hereIn this lab, you practiced your knowledge on sets, such as common set operations, the use of Venn Diagrams, the inclusion exclusion principle, and how to use sets in Python!