- The year consists of 12 distinct months, each equally likely to be chosen.
- The months starting with 'J' are January, June, and July (3 total).
- The months starting with 'M' are March and May (2 total).
- Summing these gives 5 favorable outcomes out of 12 total months, resulting in a probability of \(\boxed{\frac{5}{12}}\).
Explanation
- Understand the problem and sample space
There are 12 months in a calendar year: January, February, March, April, May, June, July, August, September, October, November, and December. Since each month is chosen at random, we assume each month is equally likely to be selected, so the probability of selecting any specific month is \(\frac{1}{12}\). The sample space size is \(12\).
- Identify favorable outcomes
We need to find the probability that a month starts with the letter 'J' or 'M'.
Let's list the months that start with 'J':
There are 3 such months.
Next, let's list the months that start with 'M':
There are 2 such months.
- Calculate total number of favorable outcomes
To find the total number of favorable outcomes, we combine the sets of months identified in the previous step. Since there is no month that starts with both 'J' and 'M', these sets are disjoint.
The total number of favorable outcomes is \(3 + 2 = 5\).
The months are {January, June, July, March, May}.
- Calculate the probability and conclude
The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes in the sample space.
Probability = \(\frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} = \frac{5}{12}\).
Checking the given options, \(\frac{5}{12}\) matches the fourth option.
Examples
This type of probability calculation is useful in scheduling or planning tasks. For example, if you are organizing meetings for a team and want to know the probability of a meeting falling in a month starting with 'J' or 'M' to avoid seasonal conflicts, this calculation provides the likelihood directly. If you have 12 potential months for an event, knowing that 5 of them fit your criteria means there is a \(\frac{5}{12}\) chance of picking such a month.