If A and B be two events in a sample space such that `P(A) = 3/(10) and p(B ) = 1/2 and P (A nn B ) = 1/5 ` . Find `P (A uu B).`

Options

A. `1/5`
B. `2/5`
C. `3/5`
D. `4/5`

Answer

To find \( P(A \cup B) \), 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: Write down the given probabilities
- \( P(A) = \frac{3}{10} \)
- \( P(B) = \frac{1}{2} \)
- \( P(A \cap B) = \frac{1}{5} \)

### Step 2: Substitute the values into the formula
Now, we substitute the values into the union formula:

\[ P(A \cup B) = \frac{3}{10} + \frac{1}{2} - \frac{1}{5} \]

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of the denominators (10, 2, and 5) is 10.

### Step 4: Convert each fraction to have the common denominator
- \( P(A) = \frac{3}{10} \) (already has the denominator 10)
- \( P(B) = \frac{1}{2} = \frac{5}{10} \)
- \( P(A \cap B) = \frac{1}{5} = \frac{2}{10} \)

### Step 5: Substitute the converted fractions back into the formula
Now we substitute these fractions back into the formula:

\[ P(A \cup B) = \frac{3}{10} + \frac{5}{10} - \frac{2}{10} \]

### Step 6: Perform the addition and subtraction
Now, we can perform the addition and subtraction:

\[ P(A \cup B) = \frac{3 + 5 - 2}{10} = \frac{6}{10} \]

### Step 7: Simplify the fraction
Now, we simplify \( \frac{6}{10} \):

\[ P(A \cup B) = \frac{3}{5} \]

### Final Answer
Thus, the probability \( P(A \cup B) \) is:

\[ \boxed{\frac{3}{5}} \]

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