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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 Probability distributions - 1
ToC
 - Axioms of probability
 - Conditional probability
   - Bayesian conditional probability
 - Random variables
   - Properties of discrete random variables
   - Binomial and Poisson discrete random variables
Axioms of probability
$$
0 \leq P(A) \leq 1
\
P(A) + P(\bar A) = 1
\
P(A \verb ! or ! B) = P(A) + P(B)
$$
Probability ranges from 0 to 1. The sum of P(A) and the opposite of A occuring is 1. For mutually exclusive events A and B, the Probability of either A or B ocurring is sum of their probabilities.
Mutually exclusive
Step1:  Conditional Probability
Generally, conditional probability is more helpful in explaining a situtation than general probabilities.
Given two events A and B with non zero probabilities, then the probability of A occurring, given that B has occurs is
$$
P(A|B) = \frac{P(A \cap B)}{P(B)}
$$
and
$$
P(B|A) = \frac{P(A \cap B)}{P(A)}
$$
The $P(A/B)$ probability of A given that B occurs, is the probability of A and B occurring $P(A \cap B)$ to the probability of B occurring $P(B)$. Thus if A and B are mutually exclusive, then there is no conditional probability.
Example
Consider the case of insurance fraud. In table below, you are given insurance type and what rate of them are fraud claims.
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Python Code:
import matplotlib.pyplot as plt
%matplotlib inline
from matplotlib_venn import venn2
venn2(subsets = (0.45, 0.15, 0.05), set_labels = ('A', 'B'))
Explanation: Probability distributions - 1
ToC
 - Axioms of probability
 - Conditional probability
   - Bayesian conditional probability
 - Random variables
   - Properties of discrete random variables
   - Binomial and Poisson discrete random variables
Axioms of probability
$$
0 \leq P(A) \leq 1
\
P(A) + P(\bar A) = 1
\
P(A \verb ! or ! B) = P(A) + P(B)
$$
Probability ranges from 0 to 1. The sum of P(A) and the opposite of A occuring is 1. For mutually exclusive events A and B, the Probability of either A or B ocurring is sum of their probabilities.
Mutually exclusive: Two events are considered mutually exclusive, if when an event is performed once, the occurrence of one of the events excludes the possibility of another.
For two independent events, probabilities of their union and intersection can be represented as
$$
P(A \cup B) = P(A) + P(B) - P(A \cap B)
$$
If A and B are mutually exclusive, then $P(A \cap B) = 0$
The reason we negate P(A intersection B) can be seen from the venn diagram below. Probabilities of A and B are (0.5 and 0.2). The probability of both A and B ocurring is 0.05. Thus to not double count the intersection, we negate it.
End of explanation
import pandas as pd
df = pd.DataFrame([[6,1,3,'Fradulent'],[14,29,47,'Not Fradulent']], 
                  columns=['Fire', 'Auto','Other','Status'])
df
Explanation: Conditional Probability
Generally, conditional probability is more helpful in explaining a situtation than general probabilities.
Given two events A and B with non zero probabilities, then the probability of A occurring, given that B has occurs is
$$
P(A|B) = \frac{P(A \cap B)}{P(B)}
$$
and
$$
P(B|A) = \frac{P(A \cap B)}{P(A)}
$$
The $P(A/B)$ probability of A given that B occurs, is the probability of A and B occurring $P(A \cap B)$ to the probability of B occurring $P(B)$. Thus if A and B are mutually exclusive, then there is no conditional probability.
Example
Consider the case of insurance fraud. In table below, you are given insurance type and what rate of them are fraud claims.
End of explanation
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