Five cards- the ten, jack, queen, king and ace of diamonds, are well shuffled with their faces downwards. One card is then picked up at random.
(a) What is the probability that the drawn card is the queen?
(b) If the queen is drawn and put aside and a second card is drawn, find the probability that the second card is (i) an ace (ii) a queen.

Answer

Let's solve the problem step by step.

### Step 1: Understand the Total Number of Cards
We have a total of 5 cards: the 10, Jack, Queen, King, and Ace of Diamonds.

### Step 2: Calculate the Probability of Drawing the Queen
To find the probability of drawing the Queen, we need to identify the number of favorable outcomes and the total number of outcomes.

- **Favorable outcomes**: There is 1 Queen.
- **Total outcomes**: There are 5 cards in total.

The probability \( P \) of drawing the Queen is given by the formula:
\[ P(\text{Queen}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{5} \]

### Step 3: Remove the Queen and Analyze the Remaining Cards
After drawing the Queen, we put it aside. Now we have the following cards left: 10, Jack, King, and Ace. This means we now have 4 cards remaining.

### Step 4: Calculate the Probability of Drawing an Ace as the Second Card
Now, we need to find the probability that the second card drawn is an Ace.

- **Favorable outcomes**: There is 1 Ace left.
- **Total outcomes**: There are now 4 cards remaining.

The probability \( P \) of drawing an Ace is given by:
\[ P(\text{Ace}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{1}{4} \]

### Step 5: Calculate the Probability of Drawing a Queen as the Second Card
Next, we need to find the probability that the second card drawn is a Queen.

- **Favorable outcomes**: There are 0 Queens left (since we already drew the Queen).
- **Total outcomes**: There are still 4 cards remaining.

The probability \( P \) of drawing a Queen is given by:
\[ P(\text{Queen}) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{0}{4} = 0 \]

### Final Answers
(a) The probability that the drawn card is the Queen is \( \frac{1}{5} \).

(b)
(i) The probability that the second card is an Ace is \( \frac{1}{4} \).
(ii) The probability that the second card is a Queen is \( 0 \).