2. Probability

Permutations and Combinations

• Permutations give the possible selections when order is important
• Combinations give the possible selections when order is not important

Permutations (Without repetition)

# Permutations Code (Without repetition)

import math

n = 4

k = 2

# Determine permutations and print result

Permutations = math.factorial(n) / math.factorial(n-k)

print("Permutations: ", Permutations)

Permutations: 12.0

Permutations (With repetition)

# Permutations Code (With repetition)

import math

n = 4

k = 2

# Determine permutations and print result

Permutations = n ** k

print("Permutations: ", Permutations)

# Permutations Code (With repetition - Alternative)

def permutation_with_repetition(n,r):

  return n ** r

permutation_with_repetition(4,2)

Permutations: 16

Combinations (Without repetition)

# Combinations Code (Without repetition)

import math

n = 4

k = 2

# Determine Permutations

Permutations = math.factorial(n) / math.factorial(n-k)

# Determine Combinations and print result

Combinations = Permutations / math.factorial(k)

print("Combinations: ", Combinations)

Combinations: 6.0

Combinations (With repetition)

# Combinations Code (With repetition)

n = 4

k = 2

# Determine Permutations

Combinations = math.factorial(n+k-1)  / (math.factorial(k) * (math.factorial(n-1)))

print("Combinations: ", Combinations)

Combinations: 10.0

Probability (Single Event)

• P(Event) = (# of outcome of event) / (# of outcomes in sample space)
• P(A) + P(B) = 1
  • P(B) = 1 - P(A)

Probability (Multiple Events)

• P (A and B) -> The intersection
• P (A or B) -> P(A) + P(B) - P (A and B)
• Mutually Exclusive Events (Could not happen at the same time)
  • P (A and B) = 0

Conditional Probability

• The probability that event A occurred given that it is known that event B has occurred is "the conditional probability of A given B"
• Dependent Event
  • P (A | B) = P (A and B) / P (B)
• Independent Event
  • P (A | B) = P (A)

Bayes' Theorem

• Provides a model for learning from observation, how to update the probability of hypotheses when given evidence
• Given a hypothesis H and evidence E
• P (H | E) = P(E | H).P(H) / P(E)
  • P (H) probability of the hypothesis before getting evidence evidence
  • P (H | E) probability of hypothesis after getting evidence