S = {1, 2, 3, ........., 30), A = {x: x is multiple of 7}, B = { x: x is multiple of 5}, C = {x:x is a multiple of 3}. If x is a member of S chosen at random find the probability that (iii) ` Acap barC`

Answer

To solve the problem, we need to find the probability that a randomly chosen member \( x \) from the set \( S \) belongs to the set \( A \cap \bar{C} \). Let's break down the steps to find this probability.

### Step 1: Define the Sets
1. **Sample Space \( S \)**: This is the set of integers from 1 to 30.
\[ S = \{1, 2, 3, \ldots, 30\} \]
The total number of elements in \( S \) is \( n(S) = 30 \).

2. **Set \( A \)**: This set contains multiples of 7 within \( S \).
\[ A = \{7, 14, 21, 28\} \]
The number of elements in \( A \) is \( n(A) = 4 \).

3. **Set \( B \)**: This set contains multiples of 5 within \( S \).
\[ B = \{5, 10, 15, 20, 25, 30\} \]
The number of elements in \( B \) is \( n(B) = 6 \).

4. **Set \( C \)**: This set contains multiples of 3 within \( S \).
\[ C = \{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\} \]
The number of elements in \( C \) is \( n(C) = 10 \).

### Step 2: Find \( A \cap C \)
Next, we need to find the intersection of sets \( A \) and \( C \) (i.e., \( A \cap C \)), which consists of elements that are both multiples of 7 and multiples of 3.

- The multiples of 7 in \( A \): \( 7, 14, 21, 28 \)
- The multiples of 3 in \( C \): \( 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 \)

The common element in both sets is:
\[ A \cap C = \{21\} \]
Thus, \( n(A \cap C) = 1 \).

### Step 3: Find \( A \cap \bar{C} \)
Now, we need to find \( A \cap \bar{C} \). The set \( \bar{C} \) contains all elements in \( S \) that are not in \( C \).

- Elements in \( \bar{C} \): All elements in \( S \) except those in \( C \).

To find \( A \cap \bar{C} \), we need to exclude the elements of \( A \) that are also in \( C \).

From \( A = \{7, 14, 21, 28\} \) and \( A \cap C = \{21\} \), we find:
\[ A \cap \bar{C} = A - (A \cap C) = \{7, 14, 28\} \]
Thus, \( n(A \cap \bar{C}) = 3 \).

### Step 4: Calculate the Probability
The probability that a randomly chosen member \( x \) from \( S \) belongs to \( A \cap \bar{C} \) is given by:
\[ P(A \cap \bar{C}) = \frac{n(A \cap \bar{C})}{n(S)} = \frac{3}{30} = \frac{1}{10} \]

### Final Answer
The probability that \( x \) is a member of \( A \cap \bar{C} \) is:
\[ \frac{1}{10} \]