Answer :
To determine the probability that the first card drawn from a standard 52-card deck is not a club and the second card drawn (after replacing the first card) is a diamond, we can follow these steps:
1. Total Number of Cards:
There are 52 cards in a standard deck.
2. Number of Clubs:
There are 13 clubs in the deck.
3. Probability that the First Card is Not a Club:
Since there are 13 clubs, there are [tex]\(52 - 13 = 39\)[/tex] cards that are not clubs. The probability that the first card drawn is not a club is calculated by:
[tex]\[ P(\text{Not a Club}) = \frac{\text{Number of non-club cards}}{\text{Total number of cards}} = \frac{39}{52} = 0.75 \][/tex]
4. Number of Diamonds:
There are 13 diamonds in the deck.
5. Probability that the Second Card is a Diamond:
The deck is replaced, maintaining the total number of cards at 52. Thus, the probability that the second card drawn is a diamond is:
[tex]\[ P(\text{Diamond}) = \frac{\text{Number of diamonds}}{\text{Total number of cards}} = \frac{13}{52} = 0.25 \][/tex]
6. Total Probability of Both Events Happening:
Since the drawing of the first card and the second card are independent events (because the card is replaced), the total probability is the product of the individual probabilities:
[tex]\[ P(\text{Not a Club and then a Diamond}) = P(\text{Not a Club}) \times P(\text{Diamond}) = 0.75 \times 0.25 = 0.1875 \][/tex]
7. Converting the Probability to a Fraction:
The decimal 0.1875 can be converted to a fraction:
[tex]\[ 0.1875 = \frac{1875}{10000} = \frac{3}{16} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{3}{16}} \][/tex]
1. Total Number of Cards:
There are 52 cards in a standard deck.
2. Number of Clubs:
There are 13 clubs in the deck.
3. Probability that the First Card is Not a Club:
Since there are 13 clubs, there are [tex]\(52 - 13 = 39\)[/tex] cards that are not clubs. The probability that the first card drawn is not a club is calculated by:
[tex]\[ P(\text{Not a Club}) = \frac{\text{Number of non-club cards}}{\text{Total number of cards}} = \frac{39}{52} = 0.75 \][/tex]
4. Number of Diamonds:
There are 13 diamonds in the deck.
5. Probability that the Second Card is a Diamond:
The deck is replaced, maintaining the total number of cards at 52. Thus, the probability that the second card drawn is a diamond is:
[tex]\[ P(\text{Diamond}) = \frac{\text{Number of diamonds}}{\text{Total number of cards}} = \frac{13}{52} = 0.25 \][/tex]
6. Total Probability of Both Events Happening:
Since the drawing of the first card and the second card are independent events (because the card is replaced), the total probability is the product of the individual probabilities:
[tex]\[ P(\text{Not a Club and then a Diamond}) = P(\text{Not a Club}) \times P(\text{Diamond}) = 0.75 \times 0.25 = 0.1875 \][/tex]
7. Converting the Probability to a Fraction:
The decimal 0.1875 can be converted to a fraction:
[tex]\[ 0.1875 = \frac{1875}{10000} = \frac{3}{16} \][/tex]
Thus, the correct answer is:
[tex]\[ \boxed{\frac{3}{16}} \][/tex]
