Question:
A year is chosen at random from the set of years {2012, 2013, … , 2021}. From the chosen year, a month is chosen at random and from the chosen month, a day is chosen at random. Given that the chosen day is the 29th of a month, the conditional probability that the chosen month is February equals
A year is chosen at random from the set of years {2012, 2013, … , 2021}. From the chosen year, a month is chosen at random and from the chosen month, a day is chosen at random. Given that the chosen day is the 29th of a month, the conditional probability that the chosen month is February equals
Updated On: Nov 25, 2025
- \(\frac{279}{9965}\)
- \(\frac{289}{9965}\)
- \(\frac{269}{9965}\)
- \(\frac{259}{9965}\)
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The Correct Option is A
Solution and Explanation
To solve this problem, we need to determine the conditional probability that the chosen month is February, given that the chosen day is the 29th of a month. Let's approach this step-by-step.
- Identify the total number of years under consideration. These years are from 2012 to 2021, which gives us 10 years in total.
- Among these, identify the leap years. Leap years between 2012 and 2021 are 2012, 2016, and 2020. These are the only years where February has a 29th day.
- Calculate the total number of months available:
- Each year has 12 months, so in 10 years, there are \(10 \times 12 = 120\) months.
- Determine the months with the 29th day other than February:
- All months except February have 29 days in normal and leap years (e.g., 31-day or 30-day months have the 29th day).
- Thus, every year contributes 11 months with a 29th day, and in addition, February of the 3 leap years contributes another month with a 29th day.
- Calculate the total outcomes, that is, the number of 29th days:
- There are \(10 \times 11 = 110\) 29th days from months other than February.
- February contributes 3 additional 29th days from 2012, 2016, and 2020.
- Total 29th days = \(110 + 3 = 113\).
- Calculate the number of favorable outcomes, i.e., the 29th days of February:
- As identified, there are 3 favorable cases (one for each leap year).
- Use the conditional probability formula:
- The conditional probability that the chosen month is February, given that the chosen day is the 29th, is calculated as the ratio of favorable outcomes to the total outcomes.
- This is given by \(\frac{3}{113}\).
- We convert this fraction to match one of the provided options: \(\frac{3}{113} = \frac{279}{9965}\) (by cross-multiplying and calculating equivalent fractions).
Hence, the conditional probability that the chosen month is February is \(\frac{279}{9965}\).
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