The probability of a month of January having 5 Sundays is ..........

Options

A. `2/7`
B. `3/7`
C. `5/7`
D. `1/7`

Answer

To find the probability of a month of January having 5 Sundays, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Total Days in January**:
January has a total of 31 days.

2. **Determine the Number of Weeks in 28 Days**:
In the first 28 days of January, there are exactly 4 complete weeks (since 28 days ÷ 7 days/week = 4 weeks). This means that each day of the week (Sunday, Monday, Tuesday, etc.) occurs exactly 4 times in these 28 days.

3. **Analyze the Remaining Days**:
After the first 28 days, there are 3 days left in January (29th, 30th, and 31st). The occurrence of Sundays in these last 3 days will determine if there are 5 Sundays in total for the month.

4. **Possible Scenarios for the Last 3 Days**:
The last 3 days can start on any day of the week. We need to check how many of these days can be Sundays:
- If January 29th is a Sunday, then there will be 5 Sundays (29th, 5th, 12th, 19th, 26th).
- If January 30th is a Sunday, then there will also be 5 Sundays (30th, 5th, 12th, 19th, 26th).
- If January 31st is a Sunday, again there will be 5 Sundays (31st, 5th, 12th, 19th, 26th).

5. **Count the Favorable Outcomes**:
There are 3 favorable outcomes (29th, 30th, or 31st being a Sunday) that lead to a total of 5 Sundays.

6. **Calculate Total Possible Outcomes**:
The total number of possible outcomes for the last 3 days is 7 (the days of the week: Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday).

7. **Calculate the Probability**:
The probability of having 5 Sundays in January can be calculated using the formula:
\[ P(\text{5 Sundays}) = \frac{\text{Number of favorable outcomes}}{\text{Total possible outcomes}} = \frac{3}{7} \]

### Final Answer:
The probability of a month of January having 5 Sundays is \(\frac{3}{7}\).

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