If `(1+4p)/(4),(1-p)/(3) and (1-2p)/(2)` are the probabilities of three mutually exclusive events then values of p may be

Options

A. `(1)/(2)`
B. `(1)/(3)`
C. `(1)/(4)`
D. none of these

Answer

To solve the problem, we need to ensure that the given expressions for probabilities are valid, meaning they must lie between 0 and 1. We will analyze each expression separately and then combine the results.

### Step 1: Analyze the first probability expression
The first probability is given by:
\[ \frac{1 + 4p}{4} \]
We need to ensure:
\[ 0 \leq \frac{1 + 4p}{4} \leq 1 \]

**Inequality 1:**
\[ \frac{1 + 4p}{4} \geq 0 \]
Multiplying both sides by 4:
\[ 1 + 4p \geq 0 \implies 4p \geq -1 \implies p \geq -\frac{1}{4} \]

**Inequality 2:**
\[ \frac{1 + 4p}{4} \leq 1 \]
Multiplying both sides by 4:
\[ 1 + 4p \leq 4 \implies 4p \leq 3 \implies p \leq \frac{3}{4} \]

So from the first expression, we have:
\[ -\frac{1}{4} \leq p \leq \frac{3}{4} \]

### Step 2: Analyze the second probability expression
The second probability is given by:
\[ \frac{1 - p}{3} \]
We need to ensure:
\[ 0 \leq \frac{1 - p}{3} \leq 1 \]

**Inequality 1:**
\[ \frac{1 - p}{3} \geq 0 \]
Multiplying both sides by 3:
\[ 1 - p \geq 0 \implies p \leq 1 \]

**Inequality 2:**
\[ \frac{1 - p}{3} \leq 1 \]
Multiplying both sides by 3:
\[ 1 - p \leq 3 \implies -p \leq 2 \implies p \geq -2 \]

So from the second expression, we have:
\[ -2 \leq p \leq 1 \]

### Step 3: Analyze the third probability expression
The third probability is given by:
\[ \frac{1 - 2p}{2} \]
We need to ensure:
\[ 0 \leq \frac{1 - 2p}{2} \leq 1 \]

**Inequality 1:**
\[ \frac{1 - 2p}{2} \geq 0 \]
Multiplying both sides by 2:
\[ 1 - 2p \geq 0 \implies 2p \leq 1 \implies p \leq \frac{1}{2} \]

**Inequality 2:**
\[ \frac{1 - 2p}{2} \leq 1 \]
Multiplying both sides by 2:
\[ 1 - 2p \leq 2 \implies -2p \leq 1 \implies p \geq -\frac{1}{2} \]

So from the third expression, we have:
\[ -\frac{1}{2} \leq p \leq \frac{1}{2} \]

### Step 4: Combine the ranges
Now we have the ranges from each expression:
1. From the first expression: \(-\frac{1}{4} \leq p \leq \frac{3}{4}\)
2. From the second expression: \(-2 \leq p \leq 1\)
3. From the third expression: \(-\frac{1}{2} \leq p \leq \frac{1}{2}\)

To find the overall valid range for \(p\), we take the intersection of these ranges:
- The lower bound is \(-\frac{1}{4}\) (the highest lower bound).
- The upper bound is \(\frac{1}{2}\) (the lowest upper bound).

Thus, the combined range for \(p\) is:
\[ -\frac{1}{4} \leq p \leq \frac{1}{2} \]

### Final Result
The values of \(p\) that satisfy all conditions are:
\[ p \in \left[-\frac{1}{4}, \frac{1}{2}\right] \]