On an average, rain falls on 12 days in every 30 days. The probability that rain will fall on just 3 days of a given week is

Options

A. `35((1)/(5))^(3)((3)/(5))^(4)`
B. `35((2)/(5))^(3)((3)/(5))^(4)`
C. `35((1)/(5))^(3)((2)/(5))^(4)`
D. `35((1)/(5))^(2)((3)/(5))^(4)`

Answer

To solve the problem, we will use the binomial distribution formula. Here’s a step-by-step solution:

### Step 1: Define the Probability of Success
The problem states that on average, rain falls on 12 days out of 30 days. Therefore, the probability \( P \) that it rains on any given day is:

\[ P = \frac{12}{30} = \frac{2}{5} \]

### Step 2: Define the Probability of Failure
The probability of not raining on a given day (failure) is:

\[ Q = 1 - P = 1 - \frac{2}{5} = \frac{3}{5} \]

### Step 3: Define the Number of Trials
Since we are looking at a week, which consists of 7 days, we have:

\[ n = 7 \]

### Step 4: Define the Number of Successes
We want to find the probability that it rains on exactly 3 days out of the 7 days. Thus, we have:

\[ k = 3 \]

### Step 5: Use the Binomial Probability Formula
The binomial probability formula is given by:

\[ P(X = k) = \binom{n}{k} P^k Q^{n-k} \]

Substituting the known values into the formula:

\[ P(X = 3) = \binom{7}{3} \left(\frac{2}{5}\right)^3 \left(\frac{3}{5}\right)^{7-3} \]

### Step 6: Calculate the Binomial Coefficient
The binomial coefficient \( \binom{7}{3} \) can be calculated as:

\[ \binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35 \]

### Step 7: Substitute and Calculate the Probability
Now we substitute back into the equation:

\[ P(X = 3) = 35 \left(\frac{2}{5}\right)^3 \left(\frac{3}{5}\right)^4 \]

Calculating \( \left(\frac{2}{5}\right)^3 \) and \( \left(\frac{3}{5}\right)^4 \):

\[ \left(\frac{2}{5}\right)^3 = \frac{8}{125} \]
\[ \left(\frac{3}{5}\right)^4 = \frac{81}{625} \]

Now substituting these values:

\[ P(X = 3) = 35 \times \frac{8}{125} \times \frac{81}{625} \]

### Step 8: Final Calculation
Now we multiply:

\[ P(X = 3) = 35 \times \frac{8 \times 81}{125 \times 625} \]

Calculating \( 8 \times 81 = 648 \):

\[ P(X = 3) = 35 \times \frac{648}{78125} \]

Finally, calculate \( 35 \times 648 \):

\[ P(X = 3) = \frac{22680}{78125} \]

### Conclusion
Thus, the probability that rain will fall on exactly 3 days of a given week is:

\[ P(X = 3) = \frac{22680}{78125} \]