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 10.

Answer

To solve the problem step by step, we will first define the sample space and then calculate the probability that neither card drawn is a 10.

### Step 1: Define the Sample Space
We have two piles of cards: one pile contains cards from the spades suit and the other from the hearts suit. Each pile contains the following cards:
- Ace (A)
- King (K)
- Queen (Q)
- Jack (J)
- Ten (10)

Thus, the sample space (S) can be represented as pairs of cards drawn from each pile.

### Step 2: List All Possible Outcomes
To form the sample space, we will pair each card from the spades pile with each card from the hearts pile. The pairs can be represented as follows:

- From Spades (S): A, K, Q, J, 10
- From Hearts (H): A, K, Q, J, 10

The sample space can be represented as:
- (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: Identify Favorable Outcomes
Next, we need to find the outcomes where neither card is a 10. We will exclude any pairs that contain the card 10.

The favorable outcomes can be listed as:
- (A, A), (A, K), (A, Q), (A, J)
- (K, A), (K, K), (K, Q), (K, J)
- (Q, A), (Q, K), (Q, Q), (Q, J)
- (J, A), (J, K), (J, Q), (J, J)

Counting these, we have:
- 4 outcomes from A
- 4 outcomes from K
- 4 outcomes from Q
- 4 outcomes from J

Thus, the total number of favorable outcomes is \(4 + 4 + 4 + 4 = 16\).

### Step 4: Calculate the Probability
The probability (P) that neither card drawn is a 10 is given by the formula:
\[ P(\text{neither card is a 10}) = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{16}{25} \]

### Final Answer
The probability that neither card will be a 10 is \(\frac{16}{25}\).

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