To solve the problem of finding the probability of drawing a black face card and then a diamond when two cards are drawn without replacement from a standard deck of cards, we can follow these steps:
1. Understand the Composition of the deck:
- A standard deck has 52 cards.
- There are 2 black face cards (Jack, Queen, King) in both spades and clubs, totaling 6 black face cards.
- There are 13 diamonds in the deck.
2. Probability of drawing a black face card first:
- There are 6 black face cards out of the total 52 cards.
- The probability of drawing a black face card first is:
[tex]\[
\frac{\text{Number of black face cards}}{\text{Total number of cards}} = \frac{6}{52} = \frac{3}{26} \approx 0.11538461538461539
\][/tex]
3. Probability of drawing a diamond next:
- After drawing the first black face card, there are now 51 cards left in the deck.
- The number of diamonds in the deck stays the same at 13.
- The probability of drawing a diamond after one card has already been drawn is:
[tex]\[
\frac{\text{Number of diamonds}}{\text{Remaining number of cards}} = \frac{13}{51} \approx 0.2549019607843137
\][/tex]
4. Combined Probability for both events to happen:
- The probability of drawing a black face card first and then a diamond (since the events are sequential and without replacement) is the product of both individual probabilities:
[tex]\[
\left( \frac{3}{26} \right) \times \left( \frac{13}{51} \right) = \frac{3 \times 13}{26 \times 51} = \frac{39}{1326} = \frac{1}{34} \approx 0.029411764705882353
\][/tex]
Hence, the probability of drawing a black face card and then a diamond when two cards are drawn without replacement is:
[tex]\[
\boxed{\frac{1}{34}}
\][/tex]
