If the probability that it will rain sometime on Monday is `(1)/(3)` and the independent probability that it will rain sometime on Tuesday is `(1)/(2)`, what is the probability that it will rain on both days ?

Options

A. `(1)/(6)`
B. `(1)/(5)`
C. `(1)/(3)`
D. `(2)/(5)`

Explanation

To find the probability that it will rain on both Monday and Tuesday, we can follow these steps:

### Step 1: Identify the probabilities
We are given:
- The probability of rain on Monday, \( P(M) = \frac{1}{3} \)
- The probability of rain on Tuesday, \( P(T) = \frac{1}{2} \)

### Step 2: Understand the independence of events
Since the events (rain on Monday and rain on Tuesday) are independent, the probability of both events occurring together is the product of their individual probabilities.

### Step 3: Calculate the probability of both events
Using the formula for independent events:
\[ P(M \text{ and } T) = P(M) \times P(T) \]
Substituting the values we have:
\[ P(M \text{ and } T) = \frac{1}{3} \times \frac{1}{2} \]

### Step 4: Perform the multiplication
Calculating the product:
\[ P(M \text{ and } T) = \frac{1 \times 1}{3 \times 2} = \frac{1}{6} \]

### Step 5: State the final answer
The probability that it will rain on both days is:
\[ \frac{1}{6} \]

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