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In a standard deck of cards, there are 13 spades, 13 clubs, 13 hearts, and 13 diamonds. The spades and the clubs are black, and the hearts and the diamonds are red.

If two cards are chosen at random from a deck, one at a time, and replaced after each pick, what is the probability that a black card is chosen first and a heart is chosen second?

A. [tex]$\frac{1}{8}$[/tex]
B. [tex]$\frac{1}{2}$[/tex]
C. [tex]$\frac{2}{3}$[/tex]
D. [tex]$\frac{3}{4}$[/tex]

Sagot :

GinnyAnswer
Let's break down the problem step by step to find the probability that a black card is chosen first and a heart is chosen second when two cards are chosen at random from a deck, one at a time, and replaced after each pick.

### Step 1: Total Number of Cards
A standard deck of cards has:
- 13 spades (black)
- 13 clubs (black)
- 13 hearts (red)
- 13 diamonds (red)

This totals to 52 cards in the deck.

### Step 2: Probability of Drawing a Black Card First
The deck has 13 spades and 13 clubs, making a total of 26 black cards. The probability of drawing a black card first is the ratio of black cards to the total number of cards.

[tex]\[ \text{Probability (Black Card First)} = \frac{26}{52} = \frac{1}{2} \][/tex]

### Step 3: Probability of Drawing a Heart Second
Since the cards are replaced after each pick, the composition of the deck remains the same. There are 13 hearts in the deck. The probability of drawing a heart second is the ratio of hearts to the total number of cards.

[tex]\[ \text{Probability (Heart Second)} = \frac{13}{52} = \frac{1}{4} \][/tex]

### Step 4: Combined Probability
Since the two events (drawing a black card first and drawing a heart second) are independent (because of replacement), we multiply the probabilities of each event to get the combined probability:

[tex]\[ \text{Combined Probability} = \text{Probability (Black Card First)} \times \text{Probability (Heart Second)} \][/tex]

[tex]\[ \text{Combined Probability} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \][/tex]

Thus, the probability that a black card is chosen first and a heart is chosen second is:

[tex]\[ \boxed{\frac{1}{8}} \][/tex]
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