Question

select the correct answer from each drop-down menu.

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

the probability that treys card will be a heart or a black-suited card is 3/4, because the set of hearts and the set of black-suited cards are choose an answer sets.

mutually exclusive
not mutually exclusive

Explanation:

Identify the given events

Using the Mutually Exclusive Events and Probability of Union knowledge points
Let \(H\) be the event of selecting a heart, and \(B\) be the event of selecting a black-suited card (spades or clubs).
A standard deck has 52 cards: 13 hearts (red), 13 diamonds (red), 13 spades (black), and 13 clubs (black).
Thus, \(P(H) = \frac{13}{52} = \frac{1}{4}\) and \(P(B) = \frac{26}{52} = \frac{1}{2}\).

Determine if the events are mutually exclusive

Using the Mutually Exclusive Events knowledge point
Since a card cannot be both a heart (which is red) and a black-suited card, the intersection is empty:

$$ H \cap B = \emptyset $$

Therefore, the set of hearts and the set of black-suited cards are mutually exclusive sets.

Calculate the probability of the union

Using the Probability of Union knowledge point
Since the events are mutually exclusive, the probability of selecting a heart or a black-suited card is:

$$ P(H \cup B) = P(H) + P(B) = \frac{13}{52} + \frac{26}{52} = \frac{39}{52} = \frac{3}{4} $$

Answer:

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