Section 1 : Probability part 3
 

[non-mutually exclusive events][cards][dice][dominoes]

 

 

 

Non-Mutually Exclusive Events(Compatible)

Two or more 'non-mutually exclusive' events can occur at the same time. In every case one event does not prevent the other happening.

The probability of either one OR both events occuring is:

P(A or B) = P(A) + P(B) − P(A ∩ B)

where P(A ∩ B) is the probability of event AND event   happening at the same time.

 

Example #1

What is the probability of getting a black card OR an Ace by drawing one card from a 52 deck.

P(Black) = 26/52

P(Ace) = 4/52

P(Black ∩ Ace) = 2/52
i.e. Black and Ace, the Ace of Clubs & the Ace of Spades

 

P(Black or Ace) = P(Black) + P(Ace) - P(Black ∩ Ace)

P(Black or Ace) = 26/52 + 4/52 - 2/52

P(Black or Ace) = 28/52 = 7/13

 

A card can either be Black or Ace or both (i.e. a Black Ace).

So that's why we need to subtract the probability of a card being both Black AND Ace .

This card has already been accounted for in the probability of the card being Black AND the probability of the card being Ace.

 

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Example #2

For a six sided die, what is the probability of getting an even number OR a number greater than 3 ?

Both events, A and B, occur when either a 4 or 6 are thrown:

A - the event of getting an even number (2, 4, 6)

B - the event of getting a number greater than 3 (4, 5, 6)

P(A) = 3/6

P(B) = 3/6

P(A ∩ B) = 2/6

 

P(A) or P(B) = P(A) + P(B) - P(A ∩ B)

P(A) or P(B) = 3/6 + 3/6 - 2/6 = 4/6 = 2/3

 

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Example #3

For a set of dominoes, what is the probability of choosing one domino and it being a 5 or any double?

P(5) = 7/28

P(double) = 7/28

P(5 double) = 1/28

 

P(5) or P(double) = P(5) + P(double) - P(5 ∩ double)

P(5) or P(double) = 7/28 + 7/28 - 1/28 = 13/28

 

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