In a leap year find the probability of
(iv) 53 Mondays or 53 Wednesday

Answer

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

### Step 1: Understand the structure of a leap year
A leap year has 366 days. This can be expressed as:
\[ 366 = 52 \text{ weeks} + 2 \text{ extra days} \]
Since there are 7 days in a week, each day of the week will occur 52 times in 52 weeks.

### Step 2: Determine the number of occurrences of each day
In a leap year:
- Each day of the week (Monday, Tuesday, Wednesday, etc.) occurs 52 times.
- The remaining 2 days will determine whether there are 53 occurrences of a specific day.

### Step 3: Identify the possible combinations of the extra days
The 2 extra days can be any combination of the days of the week. The possible pairs of extra days are:
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

### Step 4: Determine favorable outcomes for 53 Mondays or 53 Wednesdays
To have 53 Mondays, one of the extra days must be a Monday. The combinations that include Monday are:
- Sunday and Monday
- Monday and Tuesday

To have 53 Wednesdays, one of the extra days must be a Wednesday. The combinations that include Wednesday are:
- Tuesday and Wednesday
- Wednesday and Thursday

### Step 5: Count the favorable outcomes
From the combinations:
- For 53 Mondays: 2 favorable combinations (Sunday-Monday, Monday-Tuesday)
- For 53 Wednesdays: 2 favorable combinations (Tuesday-Wednesday, Wednesday-Thursday)

However, the combination of Sunday and Monday also counts towards 53 Mondays, and Tuesday and Wednesday counts towards 53 Wednesdays. Therefore, we need to ensure we do not double count:
- Total unique favorable outcomes for 53 Mondays or 53 Wednesdays: 4 (Sunday-Monday, Monday-Tuesday, Tuesday-Wednesday, Wednesday-Thursday)

### Step 6: Calculate the total outcomes
The total number of outcomes (combinations of 2 extra days) is 7.

### Step 7: Calculate the probability
The probability of having either 53 Mondays or 53 Wednesdays is given by:
\[ \text{Probability} = \frac{\text{Number of favorable outcomes}}{\text{Total outcomes}} = \frac{4}{7} \]

Thus, the final answer is:
\[ \text{Probability of 53 Mondays or 53 Wednesdays} = \frac{4}{7} \]

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