Question

An office employs thirty people. Five of them speak Sanskrit, Hindi, English. Nine speak Sanskrit, English. Twenty speak Hindi of which twelve also speak Sanskrit. Eighteen speak English. No one speak only Sanskrit. An employee selected at random is known to speak either Hindi or English, the probability that employee speaks only two of three languages is (1) \(14 / 15\) (2) \(7 / 15\) (3) \(1 / 6\) (4) \(5 / 6\)

Short answer

If the sum of each digit of a number is divisible by 9, then, the number is divisible by 9, thus, 9 is the answer

Step-by-step solution

Define the sets

Let H be the set of people who speak Hindi, S be the set of people who speak Sanskrit, and E be the set of people who speak English. We need to identify relevant numbers from the problem. Let:

Key concepts

sets and probability

Understanding sets and probability is essential for solving problems like the one in the exercise. A set is a collection of distinct objects, which in this case are the employees who speak various combinations of languages. Probability helps us understand the likelihood of an event occurring within these sets.

For instance, in the exercise, we have three sets: H (Hindi), S (Sanskrit), and E (English). Among the thirty employees:

• Twenty speak Hindi, including twelve who also speak Sanskrit.
• Nine speak Sanskrit and English.
• No employee speaks only Sanskrit.

To find probabilities, we need to calculate the size of intersections (where sets overlap) and differences (where sets do not overlap). When combining these sets, it is crucial to count each member correctly, regardless of how many languages they speak. This preparation will enable us to proceed logically to the next steps.

language proficiency

Language proficiency plays a significant role in the employee selection probability problem. Here, we analyze the proficiency across different languages to determine the likelihood of specific language combinations. The key to understanding this concept is realizing that proficiency levels can overlap, just like sets in probability theory.

In the given problem, we need to distinguish between employees who speak one, two, or all three languages. Knowing this information is essential for calculating accurate probabilities. For instance:

• Three employees speak all three languages.
• Five employees speak Sanskrit, Hindi, and English.
• Other employees could be multilingual up to two languages but not all three.

To zoom in on the concept, we must categorize each type of language proficiency and then use these categories to section our probabilities. This helps us develop precise probabilities based on the different language combinations employees possess.

employee selection probability

The probability of selecting an employee based on their language skills is the focus of our final concept. We start by noting that the employee must speak either Hindi or English. Using this condition, we calculate the likelihood of an employee speaking only two of the three languages.

Let's break this down using the information from the exercise. First calculate the total number of employees who speak only two languages out of Hindi, Sanskrit, and English by considering the given numbers:

• Employees who speak Hindi and English, but not Sanskrit.
• Employees who speak Hindi and Sanskrit, but not English.
• Employees who speak Sanskrit and English, but not Hindi.

Finally, considering 30 employees total, we find that these 'only two language' combinations form a smaller subset. By dividing this subset's size by the size of the set of employees speaking either Hindi or English, we get the desired probability. This structured approach not only simplifies our calculation but promotes a deeper understanding of the concepts involved.