As shown above, a classic deck of cards is made up of 52 cards, 26 are black, 26 are red. Each
color is split into two suits of 13 cards each (clubs and spades are black and hearts and
diamonds are red). Each suit is split into 13 individual cards (Ace, 2-10, Jack, Queen, and King).
A card is drawn randomly from a standard 52 -card deck. Find the probability of the given
event.
Write each probability as a simplified fraction.
(a) A 2 or Spade?
(b) A Spade or Diamond?

Question

UpStudy AI · Accepted Answer

**(a) A 2 or Spade**

Let  
- $$ A $$ be the event that the card is a 2. There are 4 twos in the deck.  
- $$ B $$ be the event that the card is a spade. There are 13 spades in the deck.

Notice that the card "2 of spades" is in both events. Using the inclusion‐exclusion principle:

$$
P(A \cup B) = \frac{|A| + |B| - |A \cap B|}{52}
$$

Substituting the counts:

$$
P(A \cup B) = \frac{4 + 13 - 1}{52} = \frac{16}{52} = \frac{4}{13}
$$

**(b) A Spade or Diamond**

Let  
- $$ C $$ be the event that the card is a spade (13 cards), and  
- $$ D $$ be the event that the card is a diamond (13 cards).

Since spades and diamonds are distinct suits with no overlap:

$$
P(C \cup D) = \frac{|C| + |D|}{52} = \frac{13 + 13}{52} = \frac{26}{52} = \frac{1}{2}
$$
**(a) A 2 or Spade:** The probability is $$ \frac{4}{13} $$. **(b) A Spade or Diamond:** The probability is $$ \frac{1}{2} $$.

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