Mutually exclusive union

If we know that event A and event B are mutually exclusive, the probability of either A or B happening is the sum of their probabilities:

P(A \cup B) = P(A) + P(B) \quad \text{if A and B are mutually exclusive}

this is because the probability of both A and B happening at the same time is zero, so we don’t need to subtract it from the sum of their probabilities like we

Example: rolling a die

If we roll a die, the probability of rolling a 1 is \frac{1}{6}, and the probability of rolling a 2 is also \frac{1}{6}. Because these two events are mutually exclusive, the probability of rolling a 1 or a 2 is \frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}.

Example: drawing a card

If we draw a card from a standard deck of 52 playing cards, the probability of drawing a heart is \frac{13}{52}, and the probability of drawing a black card is \frac{26}{52}. Because these two events are mutually exclusive (it’s impossible to pick a black card and a heart, as hearts are red), the probability of drawing a heart or a black card is \frac{13}{52} + \frac{26}{52} = \frac{39}{52}.