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

Sagot :

GinnyAnswer
To determine the probability of drawing a black card first and then a heart second from a standard 52-card deck, where each card is replaced after it is drawn, we follow these steps:

1. Determine the total number of black cards and hearts:
- In a standard deck of 52 cards, there are 26 black cards (13 spades + 13 clubs).
- There are 13 hearts.

2. Calculate the probability of drawing a black card first:
- The probability of drawing a black card from the deck is given by the number of black cards divided by the total number of cards.
- [tex]\[\text{P(Black card first)} = \frac{\text{Number of black cards}}{\text{Total number of cards}} = \frac{26}{52} = \frac{1}{2}\][/tex]

3. Since the card is replaced, determine the probability of drawing a heart second:
- The probability of drawing a heart from the deck is given by the number of hearts divided by the total number of cards.
- [tex]\[\text{P(Heart second)} = \frac{\text{Number of hearts}}{\text{Total number of cards}} = \frac{13}{52} = \frac{1}{4}\][/tex]

4. Calculate the combined probability of both events happening:
- The events are independent because the cards are replaced after each pick. So, the combined probability is the product of the two individual probabilities.
- [tex]\[\text{Combined probability} = \text{P(Black card first)} \times \text{P(Heart second)} = \left(\frac{1}{2}\right) \times \left(\frac{1}{4}\right) = \frac{1}{8}\][/tex]

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

The correct answer is:
[tex]\[\boxed{\frac{1}{8}}\][/tex]
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