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 hearts.

Answer

To solve the problem, we need to follow these steps:

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

### Step 2: Draw the Sample Space
The sample space consists of all possible outcomes when one card is drawn from each pile. Since there are 5 cards in each pile, the total number of outcomes is:
- Total outcomes = Number of cards in Pile 1 × Number of cards in Pile 2 = 5 × 5 = 25.

The sample space can be represented as follows:
- From Pile 1 (Spades): A, K, Q, J, 10
- From Pile 2 (Hearts): A, K, Q, J, 10

Sample Space (S):
- (A, A), (A, K), (A, Q), (A, J), (A, 10)
- (K, A), (K, K), (K, Q), (K, J), (K, 10)
- (Q, A), (Q, K), (Q, Q), (Q, J), (Q, 10)
- (J, A), (J, K), (J, Q), (J, J), (J, 10)
- (10, A), (10, K), (10, Q), (10, J), (10, 10)

### Step 3: Count the Favorable Outcomes
We need to find the probability that both cards drawn are hearts. However, since one card is drawn from the Spades pile and the other from the Hearts pile, it is impossible for both cards to be hearts.

Thus, the number of favorable outcomes (E) where both cards are hearts is:
- Number of favorable outcomes (E) = 0.

### Step 4: Calculate the Probability
The probability (P) of an event is given by the formula:
\[ P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \]

Substituting the values we have:
\[ P(E) = \frac{0}{25} = 0 \]

### Conclusion
The probability that both cards will be hearts is **0**.