The probability of occurrence of two events A and B are `1//4` and `1//2` respectively . The probability of their simultaneous occurrence is `7/50`. Find the probability that either A or B must occur.

Options

A. `61/100`
B. `29/100`
C. `39/100`
D. `56/99`

Answer

To solve the problem step by step, we will use the formula for the probability of the union of two events.

### Given:
- Probability of event A, \( P(A) = \frac{1}{4} \)
- Probability of event B, \( P(B) = \frac{1}{2} \)
- Probability of simultaneous occurrence of A and B, \( P(A \cap B) = \frac{7}{50} \)

### We need to find:
- Probability that either A or B occurs, \( P(A \cup B) \)

### Step 1: Use the formula for the probability of the union of two events
The formula for the probability of the union of two events A and B is:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 2: Substitute the given values into the formula
Substituting the values we have:
\[ P(A \cup B) = \frac{1}{4} + \frac{1}{2} - \frac{7}{50} \]

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The denominators are 4, 2, and 50. The least common multiple (LCM) of these numbers is 100.

### Step 4: Convert each fraction to have the common denominator
- Convert \( \frac{1}{4} \) to a fraction with a denominator of 100:
\[ \frac{1}{4} = \frac{1 \times 25}{4 \times 25} = \frac{25}{100} \]
- Convert \( \frac{1}{2} \) to a fraction with a denominator of 100:
\[ \frac{1}{2} = \frac{1 \times 50}{2 \times 50} = \frac{50}{100} \]
- The fraction \( \frac{7}{50} \) can be converted as follows:
\[ \frac{7}{50} = \frac{7 \times 2}{50 \times 2} = \frac{14}{100} \]

### Step 5: Substitute back into the equation
Now we can substitute these values back into the equation:
\[ P(A \cup B) = \frac{25}{100} + \frac{50}{100} - \frac{14}{100} \]

### Step 6: Combine the fractions
Now, combine the fractions:
\[ P(A \cup B) = \frac{25 + 50 - 14}{100} = \frac{61}{100} \]

### Final Answer
Thus, the probability that either A or B occurs is:
\[ P(A \cup B) = \frac{61}{100} \]