The ace , king queen , jack and ten from both the spades and hearts suits are placed in two separate piles and one card is taken from each pile. Draw the sample space diagram and find the probability that both cards will be a pair.

Answer

To solve the problem step by step, we will first identify the sample space and then calculate the probability that both cards drawn will be a pair.

### Step 1: Identify the Cards
We have two piles of cards:
- **Pile 1 (Spades)**: Ace (A), King (K), Queen (Q), Jack (J), and Ten (10)
- **Pile 2 (Hearts)**: Ace (A), King (K), Queen (Q), Jack (J), and Ten (10)

### Step 2: Draw the Sample Space
The sample space consists of all possible pairs of cards that can be drawn from the two piles. We can represent this as a grid or table:

```
Hearts
A K Q J 10
-----------------------
A | A,A A,K A,Q A,J A,10
Spades -----------------------
K | K,A K,K K,Q K,J K,10
-----------------------
Q | Q,A Q,K Q,Q Q,J Q,10
-----------------------
J | J,A J,K J,Q J,J J,10
-----------------------
10 | 10,A 10,K 10,Q 10,J 10,10
```

### Step 3: Count Total Outcomes
From the grid, we can see that there are 5 choices from the Spades pile and 5 choices from the Hearts pile. Thus, the total number of outcomes is:
\[ \text{Total Outcomes} = 5 \times 5 = 25 \]

### Step 4: Identify Favorable Outcomes
Next, we need to identify the favorable outcomes where both drawn cards are a pair. The pairs are:
- (A, A)
- (K, K)
- (Q, Q)
- (J, J)
- (10, 10)

Thus, there are 5 favorable outcomes.

### Step 5: Calculate the Probability
The probability \( P \) that both cards will be a pair is given by the formula:
\[ P(\text{pair}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{5}{25} = \frac{1}{5} \]

### Final Answer
The probability that both cards drawn will be a pair is:
\[ \frac{1}{5} \]