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Probability (Addition Rule)
Ryan Miller
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Probability (Addition Rule), Exercises of Probability and Statistics
We use the addition rule to find the probability of the union of any two events: P(A or B) = P(A) + P(B) − P(A and B)
Typology: Exercises
2021/2022
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Probability (Addition Rule)
Ryan Miller
Disjoint Events
I (^) Two events are disjoint if they have no outcomes in common I (^) Consider rolling a six-sided die, the event of rolling a six is disjoint from the event rolling an odd number
Disjoint Events
It’s easy to visually confirm this example by looking at a simple representation of the sample space:
Disjoint Events
P (Six or Odd Number) = P (Six) + P (Odd Number) = 1 / 6 + 3 / 6 = 2 / 3
The Addition Rule
I (^) In general, P ( A or B ) = P ( A ) + P ( B ) − P ( A and B ) I (^) This is known as the addition rule I (^) In the special case where events A and B are disjoint , P ( A and B ) = 0
The Addition Rule
I (^) In general, P ( A or B ) = P ( A ) + P ( B ) − P ( A and B ) I (^) This is known as the addition rule I (^) In the special case where events A and B are disjoint , P ( A and B ) = 0 I (^) In our previous examples:
P (Six or Odd Number) = P (Six) + P (Odd Number) − P (Six and Odd Number) = 1 / 6 + 3 / 6 − 0 = 2 / 3
Venn Diagrams
I (^) Venn diagrams are frequently used as a visual aid when learning the addition and complement rules I (^) The diagram below depicts survey results where 33% of college students were in a relationship (R), 25% were involved in sports (S), and 11% were in both
Venn Diagrams - Example
- P ( R or S ) = 0_._ 22 + 0_._ 11 + 0_._ 14 = 0_._ 47 (direct calculation)
Venn Diagrams - Example
- P ( R or S ) = 0_._ 22 + 0_._ 11 + 0_._ 14 = 0_._ 47 (direct calculation)
- P ( R or S ) = 0_._ 33 + 0_._ 25 − 0_._ 11 = 0_._ 47 (addition rule)
- P ( R or S ) = 1 − P (Neither) = 1 − 0_._ 53 = 0_._ 47 (complement rule)
Conclusion
We use the addition rule to find the probability of the union of any two events:
P ( A or B ) = P ( A ) + P ( B ) − P ( A and B )
If the events are independent, we know that there intersection is zero, meaning P ( A and B ) = 0 and the union of the events is simply the sum of their individual probabilities