P(A or B) = P(A) + P(B) if A and B are mutually exclusive!

What is the probability of getting a Queen + the probability of getting a 7?

P(Queen) = 4 / 52 and P(7) = 4 / 52.

Adding P(Queen) and P(7), 4 / 52 + 4 / 52 = 8 / 52, which is the probability of P(Queen or 7)!!! You might be wondering, "Does this work all the time?" It works when the two events are mutually exclusive.

If two events (call them A and B) are mutually exclusive, then the probability of A or B occuring is...

P(A or B) = P(A) + P(B)


WARNING: This only works when the two events are mutually exclusive. We will see why in this next example:

Example Find the probability that a card selected from a standard deck is an Ace or a heart.

P(Ace) = 4 / 52
Ace of Clubs Ace of Spades Ace of Hearts Ace of Diamonds
P(heart) = 13 / 52
Ace of Hearts 2 of Hearts 3 of Hearts 4 of Hearts
5 of Hearts 6 of Hearts 7 of Hearts 8 of Hearts 9 of Hearts
10 of Hearts Jack of Hearts Queen of Hearts King of Hearts


Note that P(Ace) and P(heart) are not mutually exclusive since they share a common card - the Ace of Hearts (colored purple above). If we add P(Ace) and P(heart) to get P(Ace or heart), we will count any card that is both an Ace and a heart twice. Therefore, we must subtract out the number of cards that are both Aces and hearts.

P(Ace) + P(heart) - P(Ace and heart) = 4 / 52 + 13 / 52 - 1 / 52 = 16 / 52.

Counting above, you can see that there are 16 unique cards that are either Aces or hearts. Therefore, P(Ace or heart) = 16 / 52.


In General If A and B are two events:

P(A or B) = P(A) + P(B) - P(A and B)


Example What is the probability that a card drawn at random from a deck of 52 cards has an even number written on the card or is black?

Solution To find P(even number or black), first notice that the two events are not mutually exclusive. Using the general formula,

P(even or black) = P(even) + P(black) - P(even and black)
= 20 / 52 + 26 / 52 - 10 / 52
= 36 / 52


Now that you've learned about events that are mutually exclusive, let's take a look at another aspect of probability: conditional probability.



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