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}.

flashcards

QuestionAnswer
What is the general formula for the probability of A or B when A and B are mutually exclusive?P(A \cup B) = P(A) + P(B)
Why does the union rule for mutually exclusive events not include a subtraction term?Because the probability of both events happening at the same time is zero, so we don’t need to subtract the intersection.
In the die roll example, what is the probability of rolling a 1 or a 2?\frac{1}{6} + \frac{1}{6} = \frac{2}{6} = \frac{1}{3}
In a standard deck of 52 cards, what is the probability of drawing a heart or a black card?\frac{13}{52} + \frac{26}{52} = \frac{39}{52}
Why are “drawing a heart” and “drawing a black card” mutually exclusive?Because it is impossible to pick a black card and a heart, as hearts are red.