A card is chosen at random from a standard deck of 52 playing cards. Without replacing it, a second card is chosen. What is the probability that the first card chosen is a queen and the second card chosen is a jack?

Answer

To solve the problem of finding the probability that the first card chosen is a queen and the second card chosen is a jack, we can follow these steps:

### Step 1: Determine the total number of cards in the deck
A standard deck of playing cards contains 52 cards.

**Hint:** Remember that a standard deck consists of 4 suits, each containing 13 cards.

### Step 2: Calculate the probability of drawing a queen first
There are 4 queens in the deck (one from each suit). Therefore, the probability of drawing a queen first is:

\[ P(\text{Queen first}) = \frac{\text{Number of Queens}}{\text{Total Cards}} = \frac{4}{52} \]

**Hint:** The probability is calculated as the number of favorable outcomes divided by the total number of outcomes.

### Step 3: Determine the number of cards left after the first draw
After drawing one queen, there are now 51 cards left in the deck.

**Hint:** Always remember to adjust the total number of cards after each draw, especially when cards are not replaced.

### Step 4: Calculate the probability of drawing a jack second
There are still 4 jacks in the deck, as drawing a queen does not affect the number of jacks. Therefore, the probability of drawing a jack second is:

\[ P(\text{Jack second}) = \frac{\text{Number of Jacks}}{\text{Total Cards Left}} = \frac{4}{51} \]

**Hint:** The number of jacks remains the same unless a jack is drawn, which is not the case here.

### Step 5: Calculate the combined probability
Since the events are sequential (drawing a queen first and then a jack), we multiply the probabilities of the two independent events:

\[ P(\text{Queen first and Jack second}) = P(\text{Queen first}) \times P(\text{Jack second}) = \frac{4}{52} \times \frac{4}{51} \]

### Step 6: Simplify the expression
Now, we simplify the expression:

\[ P(\text{Queen first and Jack second}) = \frac{4 \times 4}{52 \times 51} = \frac{16}{2652} \]

We can further simplify this fraction:

\[ \frac{16}{2652} = \frac{4}{663} \]

### Final Answer
Thus, the probability that the first card chosen is a queen and the second card chosen is a jack is:

\[ \frac{4}{663} \]

---