In a leap year find the probability of
(i) 53 Mondays and 53 Tuesdays

Answer

To solve the problem of finding the probability of having 53 Mondays and 53 Tuesdays in a leap year, we can follow these steps:

### Step 1: Understand the structure of a leap year
A leap year consists of 366 days. This can be divided into 52 complete weeks and 2 extra days.

**Hint:** Remember that a week has 7 days, so 52 weeks account for 364 days.

### Step 2: Determine the number of each day in a leap year
Since there are 52 complete weeks, each day of the week (Monday, Tuesday, etc.) will occur 52 times. Therefore, in a leap year, there are:
- 52 Mondays
- 52 Tuesdays
- 52 Wednesdays
- 52 Thursdays
- 52 Fridays
- 52 Saturdays
- 52 Sundays

This accounts for 364 days. The remaining 2 days will determine if there are 53 of any particular day.

**Hint:** Think about how the extra days can affect the count of each day of the week.

### Step 3: Identify the possible combinations of the extra days
The 2 extra days can be any of the following pairs:
1. Sunday and Monday
2. Monday and Tuesday
3. Tuesday and Wednesday
4. Wednesday and Thursday
5. Thursday and Friday
6. Friday and Saturday
7. Saturday and Sunday

**Hint:** List out the pairs of days that can occur as the extra days.

### Step 4: Find the favorable outcome
To have 53 Mondays and 53 Tuesdays, one of the extra days must be a Monday and the other must be a Tuesday. The only pair that satisfies this condition is:
- Monday and Tuesday

**Hint:** Look for the specific pair that includes both required days.

### Step 5: Calculate the total outcomes
There are a total of 7 possible pairs of extra days.

**Hint:** Count how many different pairs of days can occur from the 7 days of the week.

### Step 6: Calculate the probability
The probability of having 53 Mondays and 53 Tuesdays is given by the formula:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} \]
In this case, there is 1 favorable outcome (Monday and Tuesday) and 7 total outcomes (all possible pairs of extra days).

Thus, the probability is:
\[ \text{Probability} = \frac{1}{7} \]

### Final Answer
The probability of having 53 Mondays and 53 Tuesdays in a leap year is \( \frac{1}{7} \).

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