Sets A, B, C, `Ann B , AnnC , B nn C` and `AnnB nnC` have 35, 40, 45, 13, 12, 14 and 5 elements respectively. An element is selected at random from the set `AuuBuuC`. The probability that the selected element belongs to only set A is:

Options

A. `13/86`
B. `35/86`
C. `5/86`
D. `15/86`

Answer

To solve the problem, we need to find the probability that a randomly selected element from the set \( A \cup B \cup C \) belongs only to set \( A \). We will follow these steps:

### Step 1: Calculate the total number of elements in \( A \cup B \cup C \)

The formula for the total number of elements in the union of three sets is given by:

\[ n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(A \cap C) + n(A \cap B \cap C) \]

Given values:
- \( n(A) = 35 \)
- \( n(B) = 40 \)
- \( n(C) = 45 \)
- \( n(A \cap B) = 13 \)
- \( n(B \cap C) = 12 \)
- \( n(A \cap C) = 14 \)
- \( n(A \cap B \cap C) = 5 \)

Substituting these values into the formula:

\[ n(A \cup B \cup C) = 35 + 40 + 45 - 13 - 12 - 14 + 5 \]

Calculating step-by-step:

1. Sum of individual sets: \( 35 + 40 + 45 = 120 \)
2. Sum of intersections: \( 13 + 12 + 14 = 39 \)
3. Total: \( 120 - 39 + 5 = 86 \)

Thus,

\[ n(A \cup B \cup C) = 86 \]

### Step 2: Calculate the number of elements that belong only to set \( A \)

To find the number of elements that belong only to set \( A \), we use the formula:

\[ n(\text{only } A) = n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C) \]

Substituting the known values:

\[ n(\text{only } A) = 35 - 13 - 14 + 5 \]

Calculating step-by-step:

1. Subtract intersections: \( 35 - 13 - 14 = 8 \)
2. Add the intersection of all three sets: \( 8 + 5 = 13 \)

Thus,

\[ n(\text{only } A) = 13 \]

### Step 3: Calculate the probability that a randomly selected element belongs only to set \( A \)

The probability \( P \) that a randomly selected element belongs only to set \( A \) is given by:

\[ P(\text{only } A) = \frac{n(\text{only } A)}{n(A \cup B \cup C)} \]

Substituting the values we calculated:

\[ P(\text{only } A) = \frac{13}{86} \]

### Final Answer

Thus, the probability that the selected element belongs only to set \( A \) is:

\[ \frac{13}{86} \]

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