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 neither card will be a king or jack.

Answer

To solve the problem step by step, we will first identify the cards and then determine the sample space and the probability that neither card drawn is a king or a jack.

### Step 1: Identify the Cards
We have two 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 pairs of cards that can be drawn, one from each pile. We can represent this as a grid or table where the rows represent the cards from the spades pile and the columns represent the cards from the hearts pile.

The sample space can be represented as follows:

| | A (Hearts) | K (Hearts) | Q (Hearts) | J (Hearts) | 10 (Hearts) |
|-------|------------|------------|------------|------------|-------------|
| A (Spades) | (A, A) | (A, K) | (A, Q) | (A, J) | (A, 10) |
| K (Spades) | (K, A) | (K, K) | (K, Q) | (K, J) | (K, 10) |
| Q (Spades) | (Q, A) | (Q, K) | (Q, Q) | (Q, J) | (Q, 10) |
| J (Spades) | (J, A) | (J, K) | (J, Q) | (J, J) | (J, 10) |
| 10 (Spades)| (10, A) | (10, K) | (10, Q) | (10, J) | (10, 10) |

From this table, we can see that there are a total of \(5 \times 5 = 25\) possible outcomes in the sample space.

### Step 3: Identify Favorable Outcomes
Next, we need to find the outcomes where neither card is a King (K) or a Jack (J).

The cards that are not Kings or Jacks are:
- From Spades: A, Q, 10
- From Hearts: A, Q, 10

Now we can list the favorable outcomes:

| | A (Hearts) | Q (Hearts) | 10 (Hearts) |
|-------|------------|------------|-------------|
| A (Spades) | (A, A) | (A, Q) | (A, 10) |
| Q (Spades) | (Q, A) | (Q, Q) | (Q, 10) |
| 10 (Spades)| (10, A) | (10, Q) | (10, 10) |

Counting these, we have:
- (A, A)
- (A, Q)
- (A, 10)
- (Q, A)
- (Q, Q)
- (Q, 10)
- (10, A)
- (10, Q)
- (10, 10)

This gives us a total of 9 favorable outcomes.

### Step 4: Calculate the Probability
The probability that neither card drawn is a King or a Jack is given by the ratio of the number of favorable outcomes to the total number of outcomes in the sample space.

\[ \text{Probability} = \frac{\text{Number of Favorable Outcomes}}{\text{Total Outcomes}} = \frac{9}{25} \]

### Final Answer
The probability that neither card will be a King or a Jack is \(\frac{9}{25}\).

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