Joint probability
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Joint probability refers to the likelihood of two or more events occurring simultaneously, categorized into mutually exclusive and non-mutually exclusive events. Mutually exclusive events cannot occur at the same time, leading to a probability of zero for their simultaneous occurrence, whereas non-mutually exclusive events can occur together, requiring a correction in their combined probability calculation. The document provides formulas for both types of events and examples to illustrate these concepts.
Joint probability
- 2. Joint Probability is the likelihood of more than one event occurring at the same time.
- 3. Two types of Joint Probability • Mutually Exclusive Events (without common outcomes) • Non-Mutually Exclusive Events (with common outcomes)
- 4. Mutually Exclusive Events Examples: • Turning left and turning right are Mutually Exclusive (you can't do both at the same time) • Tossing a coin: Heads and Tails are Mutually Exclusive (can't happen at the same time) • Cards: Kings and Aces are Mutually Exclusive (can’t be both)
- 5. Aces and Kings are Mutually Exclusive (can't be both)
- 6. When two events (call them "A" and "B") are Mutually Exclusive it is impossible for them to happen together: P(A and B) = 0 "The probability of A and B together equals 0 (impossible)"
- 7. But the probability of A or B is the sum of the individual probabilities: P(A or B) = P(A) + P(B) "The probability of A or B equals the probability of A plus the probability of B"
- 8. Example: A Deck of Cards In a Deck of 52 Cards: • the probability of a King is 1/13, so P(King)=1/13 • the probability of an Ace is also 1/13, so P(Ace)=1/13
- 9. When we combine those two Events: • The probability of a card being a King and an Ace is 0 (Impossible) • The probability of a card being a King or an Ace is (1/13) + (1/13) = 2/13
- 10. Which is written like this: •P(King and Ace) = 0 •P(King or Ace) = (1/13) + (1/13) = 2/13
- 11. Special Notation • Instead of "and" you will often see the symbol ∩ (which is the "Intersection" symbol used in Venn Diagrams) • Instead of "or" you will often see the symbol ∪ (the "Union" symbol)
- 12. Remembering To help you remember, think: • "Or has more ... than And" • Also ∪ is like a cup which holds more than ∩
- 13. Non Mutually Exclusive Events • Example: Hearts and Kings
- 14. • Hearts and Kings together is only the King of Hearts: • Hearts or Kings is: all the Hearts (13 of them) all the Kings (4 of them) But that counts the King of Hearts twice!
- 15. So we correct our answer, by subtracting the extra "and" part: 16 Cards = 13 Hearts + 4 Kings − the 1 extra King of Hearts
- 16. Formula: • P(A or B) = P(A) + P(B) − P(A and B) "The probability of A or B equals the probability of A plus the probability of B minus the probability of A and B" • Here is the same formula, but using ∪ and ∩: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)
- 17. Summary: • Mutually Exclusive A and B together is impossible: P(A and B) = 0 A or B is the sum of A and B: P(A or B) = P(A) + P(B) • Non-Mutually Exclusive A or B is the sum of A and B minus A and B: P(A or B) = P(A) + P(B) − P(A and B)