Answered

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 the 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]



Answer :

GinnyAnswer
To determine the probability of drawing a black card first and then a heart from a standard deck of 52 cards, where the card is replaced after each pick, we need to follow these steps:

1. Calculate the probability of drawing a black card first:
- In a standard deck, there are 52 cards in total.
- There are 26 black cards (13 spades and 13 clubs).
- The probability of drawing a black card on the first draw is given by the ratio of black cards to the total number of cards.
[tex]\[ \text{Probability of drawing a black card first} = \frac{\text{number of black cards}}{\text{total number of cards}} = \frac{26}{52} = \frac{1}{2} \][/tex]

2. Calculate the probability of drawing a heart second after replacing the first card:
- After replacing the first card, the deck returns to its original state of 52 cards.
- There are 13 hearts in the deck.
- The probability of drawing a heart on the second draw is given by the ratio of hearts to the total number of cards.
[tex]\[ \text{Probability of drawing a heart second} = \frac{\text{number of hearts}}{\text{total number of cards}} = \frac{13}{52} = \frac{1}{4} \][/tex]

3. Calculate the combined probability for both independent events:
- Drawing a black card first and then a heart are two independent events.
- To find the combined probability of both events happening, we multiply the probabilities of each event.
[tex]\[ \text{Total probability} = \text{Probability of black card first} \times \text{Probability of heart second} \][/tex]
[tex]\[ \text{Total probability} = \frac{1}{2} \times \frac{1}{4} = \frac{1}{8} \][/tex]

Therefore, the probability that a black card is chosen first and a heart is chosen second is [tex]\( \frac{1}{8} \)[/tex].

This matches the probability found earlier through the detailed step-by-step solution. Hence, the correct answer is:

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