Question

select the correct answer from each drop-down menu.

trey randomly selects one card a from a standard 52-card deck.

the probability that treys card will be a heart or a black-suited card is choose an answer, because the set of hearts and the set of black-suited cards are choose an answer sets.

Explanation:

Identify the given sets and their sizes

We are dealing with a standard 52-card deck.
Let \(H\) be the set of hearts.
Let \(B\) be the set of black-suited cards (spades and clubs).
The number of cards in each set is:

$$ n(H) = 13 $$

$$ n(B) = 26 $$

Determine if the sets are mutually exclusive

Hearts are a red suit, while spades and clubs are black suits.
There is no card that is both a heart and a black-suited card.
Therefore, the intersection of the two sets is empty:

$$ H \cap B = \emptyset $$

This means the set of hearts and the set of black-suited cards are disjoint (or mutually exclusive) sets.

Calculate the probability of the union

Since the events are mutually exclusive, the probability of selecting a heart or a black-suited card is the sum of their individual probabilities:

$$ P(H \cup B) = P(H) + P(B) = \frac{13}{52} + \frac{26}{52} = \frac{39}{52} $$

Simplifying the fraction:

$$ \frac{39}{52} = \frac{3}{4} $$

Answer:

The probability that Trey's card will be a heart or a black-suited card is <blank>\(3/4\)</blank>, because the set of hearts and the set of black-suited cards are <blank>disjoint</blank> sets.