If ` P(X) = (3)/( 10) , P(Y) = (2)/(5) and P(X cupY) = (3)/(5) , "then " P((Y)/(X)) + P ((X)/(Y))` equals

Options

A. `(1)/(4)`
B. `(1)/(3)`
C. `(5)/(12)`
D. `(7)/(12)`

Answer

To solve the problem, we need to find \( P\left(\frac{Y}{X}\right) + P\left(\frac{X}{Y}\right) \) given:

- \( P(X) = \frac{3}{10} \)
- \( P(Y) = \frac{2}{5} \)
- \( P(X \cup Y) = \frac{3}{5} \)

### Step 1: Find \( P(X \cap Y) \)

Using the formula for the probability of the union of two events:

\[ P(X \cup Y) = P(X) + P(Y) - P(X \cap Y) \]

Substituting the known values:

\[ \frac{3}{5} = \frac{3}{10} + \frac{2}{5} - P(X \cap Y) \]

### Step 2: Convert \( P(Y) \) to a common denominator

To perform the addition, we need a common denominator. The common denominator for \( \frac{3}{10} \) and \( \frac{2}{5} \) is 10.

Convert \( \frac{2}{5} \):

\[ \frac{2}{5} = \frac{4}{10} \]

### Step 3: Substitute and solve for \( P(X \cap Y) \)

Now substitute back into the equation:

\[ \frac{3}{5} = \frac{3}{10} + \frac{4}{10} - P(X \cap Y) \]

Combine the fractions on the right side:

\[ \frac{3}{5} = \frac{7}{10} - P(X \cap Y) \]

Now convert \( \frac{3}{5} \) to a fraction with a denominator of 10:

\[ \frac{3}{5} = \frac{6}{10} \]

So we have:

\[ \frac{6}{10} = \frac{7}{10} - P(X \cap Y) \]

Rearranging gives:

\[ P(X \cap Y) = \frac{7}{10} - \frac{6}{10} = \frac{1}{10} \]

### Step 4: Calculate \( P\left(\frac{Y}{X}\right) \) and \( P\left(\frac{X}{Y}\right) \)

Using the definitions of conditional probability:

\[ P\left(\frac{Y}{X}\right) = \frac{P(X \cap Y)}{P(X)} = \frac{\frac{1}{10}}{\frac{3}{10}} = \frac{1}{3} \]

\[ P\left(\frac{X}{Y}\right) = \frac{P(X \cap Y)}{P(Y)} = \frac{\frac{1}{10}}{\frac{2}{5}} = \frac{1}{10} \cdot \frac{5}{2} = \frac{1}{4} \]

### Step 5: Add the two probabilities

Now we add the two probabilities:

\[ P\left(\frac{Y}{X}\right) + P\left(\frac{X}{Y}\right) = \frac{1}{3} + \frac{1}{4} \]

### Step 6: Find a common denominator to add

The common denominator for 3 and 4 is 12:

\[ \frac{1}{3} = \frac{4}{12}, \quad \frac{1}{4} = \frac{3}{12} \]

Now add them:

\[ P\left(\frac{Y}{X}\right) + P\left(\frac{X}{Y}\right) = \frac{4}{12} + \frac{3}{12} = \frac{7}{12} \]

### Final Answer

Thus, the final answer is:

\[ P\left(\frac{Y}{X}\right) + P\left(\frac{X}{Y}\right) = \frac{7}{12} \]

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