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 kings ,

Answer

To solve the problem step-by-step, we will follow these instructions:

### Step 1: Identify the cards in each pile
We have two piles of cards:
- **Pile 1 (Spades)**: Ace, King, Queen, Jack, Ten
- **Pile 2 (Hearts)**: Ace, King, Queen, Jack, Ten

### Step 2: Draw the sample space
The sample space consists of all possible outcomes when one card is drawn from each pile. We can represent the sample space as a set of ordered pairs (X, Y), where X is a card from the Spades pile and Y is a card from the Hearts pile.

The sample space can be represented as follows:
- (Ace, Ace), (Ace, King), (Ace, Queen), (Ace, Jack), (Ace, Ten)
- (King, Ace), (King, King), (King, Queen), (King, Jack), (King, Ten)
- (Queen, Ace), (Queen, King), (Queen, Queen), (Queen, Jack), (Queen, Ten)
- (Jack, Ace), (Jack, King), (Jack, Queen), (Jack, Jack), (Jack, Ten)
- (Ten, Ace), (Ten, King), (Ten, Queen), (Ten, Jack), (Ten, Ten)

Thus, the complete sample space is:
\[ S = \{ (A, A), (A, K), (A, Q), (A, J), (A, 10), (K, A), (K, K), (K, Q), (K, J), (K, 10), (Q, A), (Q, K), (Q, Q), (Q, J), (Q, 10), (J, A), (J, K), (J, Q), (J, J), (J, 10), (10, A), (10, K), (10, Q), (10, J), (10, 10) \} \]
This gives us a total of \(5 \times 5 = 25\) possible outcomes.

### Step 3: Calculate the probability that both cards drawn are Kings
Now, we need to find the probability that both cards drawn from the piles are Kings.

1. **Favorable outcomes**: The only favorable outcome for this event is (King, King).
2. **Total outcomes**: As calculated earlier, the total number of outcomes in the sample space is 25.

The probability \(P\) of both cards being Kings is given by the formula:
\[ P(\text{Both Kings}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{1}{25} \]

### Final Answer
Thus, the probability that both cards drawn will be Kings is \(\frac{1}{25}\).

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