Probability of a Union of Events

• We also know that for a set of disjoin events A(i)
  
• For every two events

P(A U B) = P(A) + P (B) - P(A ∩ B)

• This can be extended to 3 or More events

P(A U B U C) = P(A) + P(B) + P(C) - {P (A ∩ B) + P(B ∩ C) + P(A ∩ C)} + P(A ∩ B ∩ C)

• Example: Consider 200 students
• 50 students take programming (P)
• 100 Students take electronics (E)
• 75 Students take Maths (M)
• 30 students take programming+Electronics
• 45 students take electronics+maths
• 25 students take electornics+programming
• 15 students take all the three classes
• Some students take no classes from the list
• What is the probability that a student takes atleast one class
  

Conditional Probability

• Lets assume you play outdoor sports, you care about rain
• Compare
• Probability it will rain today (Unconditional Probability)
• Probability it will rain today give its been raining for last two days (Conditional Probability)
• The conditional probability of event A given event B has occured is
  
• Example-1:
• The unconditional probability that you will roll 3 on a balanced die roll => 1/6
• But what if you observed an odd number
• This reduces number of options => {1,3,5} => conditional probability => 1/3
• Lets use the formula
  
• Scenario: We have gone to food plaza. Our friend Arundhati has placed a bet of 100 rs saying next person we meet will love both biryani and pizza
• We have collected some data and represented in colorful venn diagram
  
• So if we represent in contingency table, They contain the same information
  
• unconditional probabilities:
  
• Row total & Column Total
  
  
• What is probability of not loving biryani given he loves pizza => 5/7