If `E` and `F` are two independent events and `P(E)=1/3,P(F)=1/4,` then find `P(EuuF)`.

Explanation

To find the probability of the union of two independent events \(E\) and \(F\), we can use the formula:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

Given:
- \(P(E) = \frac{1}{3}\)
- \(P(F) = \frac{1}{4}\)

Since \(E\) and \(F\) are independent events, the probability of their intersection can be calculated as:

\[ P(E \cap F) = P(E) \times P(F) \]

Now, let's calculate \(P(E \cap F)\):

\[ P(E \cap F) = P(E) \times P(F) = \frac{1}{3} \times \frac{1}{4} = \frac{1}{12} \]

Next, we substitute the values into the union formula:

\[ P(E \cup F) = P(E) + P(F) - P(E \cap F) \]

Substituting the known values:

\[ P(E \cup F) = \frac{1}{3} + \frac{1}{4} - \frac{1}{12} \]

To perform this addition and subtraction, we need a common denominator. The least common multiple of 3, 4, and 12 is 12.

Now, we convert each fraction:

\[ P(E) = \frac{1}{3} = \frac{4}{12} \]
\[ P(F) = \frac{1}{4} = \frac{3}{12} \]
\[ P(E \cap F) = \frac{1}{12} \]

Now substituting these values back into the equation:

\[ P(E \cup F) = \frac{4}{12} + \frac{3}{12} - \frac{1}{12} \]

Now, combine the fractions:

\[ P(E \cup F) = \frac{4 + 3 - 1}{12} = \frac{6}{12} \]

Finally, simplify the fraction:

\[ P(E \cup F) = \frac{1}{2} \]

Thus, the probability of \(E \cup F\) is:

\[ \boxed{\frac{1}{2}} \]