If `P(A)=1//4,P(B)=2//5` then find the range of `P(AuuB)`

Options

A. `(1//5,13//20)`
B. `(1//4,13//20)`
C. `(2//5,13//20)`
D. None of these

Answer

To find the range of \( P(A \cup B) \) given \( P(A) = \frac{1}{4} \) and \( P(B) = \frac{2}{5} \), we can use the formula for the probability of the union of two events:

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

### Step 1: Calculate \( P(A) + P(B) \)

First, we need to calculate the sum of \( P(A) \) and \( P(B) \):

\[ P(A) + P(B) = \frac{1}{4} + \frac{2}{5} \]

To add these fractions, we need a common denominator. The least common multiple of 4 and 5 is 20.

\[ P(A) = \frac{1}{4} = \frac{5}{20} \]
\[ P(B) = \frac{2}{5} = \frac{8}{20} \]

Now, adding these:

\[ P(A) + P(B) = \frac{5}{20} + \frac{8}{20} = \frac{13}{20} \]

### Step 2: Determine the range of \( P(A \cap B) \)

The next step is to determine the possible values for \( P(A \cap B) \). The intersection of two events cannot exceed the probability of either event:

\[ P(A \cap B) \leq \min(P(A), P(B)) \]

Calculating the minimum:

\[ \min\left(\frac{1}{4}, \frac{2}{5}\right) = \frac{1}{4} \]

### Step 3: Find the maximum value of \( P(A \cup B) \)

To find the maximum value of \( P(A \cup B) \), we assume \( P(A \cap B) \) is at its minimum (which is 0):

\[ P(A \cup B)_{\text{max}} = P(A) + P(B) - 0 = \frac{13}{20} \]

### Step 4: Find the minimum value of \( P(A \cup B) \)

To find the minimum value of \( P(A \cup B) \), we assume \( P(A \cap B) \) is at its maximum (which is \( \frac{1}{4} \)):

\[ P(A \cup B)_{\text{min}} = P(A) + P(B) - P(A \cap B)_{\text{max}} = \frac{1}{4} + \frac{2}{5} - \frac{1}{4} \]

Calculating this:

\[ P(A \cup B)_{\text{min}} = \frac{1}{4} + \frac{2}{5} - \frac{1}{4} = \frac{2}{5} \]

### Step 5: Conclusion

Thus, the range of \( P(A \cup B) \) is:

\[ \left[\frac{2}{5}, \frac{13}{20}\right] \]