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 29^th of a month, the conditional probability that the chosen month is February equals

• A. \(\frac{279}{9965}\)
• B. \(\frac{289}{9965}\)
• C. \(\frac{269}{9965}\)
• D. \(\frac{259}{9965}\)

Answer 1

The Correct Option is A

To solve this problem, we need to determine the conditional probability that the chosen month is February, given that the chosen day is the 29^th of a month. Let's approach this step-by-step.

1. Identify the total number of years under consideration. These years are from 2012 to 2021, which gives us 10 years in total.
2. 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 29^th day.
3. Calculate the total number of months available:
   • Each year has 12 months, so in 10 years, there are \(10 \times 12 = 120\) months.
4. Determine the months with the 29^th 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 29^th day, and in addition, February of the 3 leap years contributes another month with a 29^th day.
5. 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 = \(110 + 3 = 113\).
6. 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).
7. Use the conditional probability formula:
   • The conditional probability that the chosen month is February, given that the chosen day is the 29^th, 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 \(\frac{279}{9965}\).