Understanding the Applicant Data and Probability

The problem provides data about 90 applicants for a job, categorized by their graduate status and years of experience. We are asked to calculate a specific conditional probability based on this data. Let's first organize the given information in a clear table.

We are given the following events:

• G: The first applicant interviewed is a graduate.
• T: The first applicant interviewed has at least 3 years experience.

The question asks for \(P(\overline T | \overline G)\), which is the probability that the first applicant interviewed has less than 3 years experience, given that they are a non-graduate. The notation \(\overline G\) means the applicant is not a graduate (i.e., a non-graduate), and \(\overline T\) means the applicant does not have at least 3 years experience (i.e., has less than 3 years experience).

Calculating Conditional Probability \(P(\overline T | \overline G)\)

Conditional probability \(P(A|B)\) is the probability of event A occurring given that event B has already occurred. It is calculated using the formula:

\(P(A|B) = \frac{P(A \cap B)}{P(B)}\)

In our case, \(A = \overline T\) and \(B = \overline G\). So, we need to calculate \(P(\overline T | \overline G) = \frac{P(\overline T \cap \overline G)}{P(\overline G)}\).

Alternatively, for problems involving selecting one item from a group, the conditional probability \(P(A|B)\) can be calculated directly as the ratio of the number of outcomes where both A and B occur to the number of outcomes where B occurs:

\(P(A|B) = \frac{\text{Number of outcomes in } A \cap B}{\text{Number of outcomes in } B}\)

Let's use the direct method based on the counts from the table.

• Event \(\overline G\) is that the applicant is a non-graduate. From the table, the total number of non-graduates is 36. This is the size of our reduced sample space for the condition \(\overline G\).
• Event \(\overline T \cap \overline G\) is that the applicant is both a non-graduate AND has less than 3 years experience. From the table, the number of non-graduates with less than 3 years experience is 27.

Now, we can calculate the conditional probability:

\(P(\overline T | \overline G) = \frac{\text{Number of non-graduates with less than 3 years experience}}{\text{Total number of non-graduates}}\)

\(P(\overline T | \overline G) = \frac{27}{36}\)

Simplifying the Probability

We can simplify the fraction \(\frac{27}{36}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 9.

\(\frac{27 \div 9}{36 \div 9} = \frac{3}{4}\)

Therefore, \(P(\overline T | \overline G)\) is equal to \(\frac{3}{4}\).

Conclusion

The probability that the first applicant interviewed has less than 3 years experience, given that they are a non-graduate, is \(\frac{3}{4}\).

Revision Table: Key Probabilities from Applicant Data

Let's quickly summarize some related total probabilities from the table for revision.

• Total Applicants = 90
• \(P(G)\) (Probability of being a graduate) = \(\frac{54}{90}\)
• \(P(\overline G)\) (Probability of being a non-graduate) = \(\frac{36}{90}\)
• \(P(T)\) (Probability of having \(\ge\) 3 yrs experience) = \(\frac{27}{90}\)
• \(P(\overline T)\) (Probability of having < 3 yrs experience) = \(\frac{63}{90}\)
• \(P(\overline T \cap \overline G)\) (Probability of being non-graduate AND having < 3 yrs experience) = \(\frac{27}{90}\)

We calculated \(P(\overline T | \overline G) = \frac{P(\overline T \cap \overline G)}{P(\overline G)} = \frac{27/90}{36/90} = \frac{27}{36} = \frac{3}{4}\), which matches our direct calculation.

Additional Information: Conditional Probability Concepts

Conditional probability is a fundamental concept in probability theory. It helps us understand how the probability of an event changes when we know that another event has occurred. The formula \(P(A|B) = \frac{P(A \cap B)}{P(B)}\) is key, provided \(P(B) \gt 0\). In simpler terms for discrete events, it's the proportion of times A occurs within the instances where B occurs.

It's different from \(P(A \cap B)\), which is the probability that both A and B happen together out of the entire sample space. Conditional probability shrinks the sample space to just the outcomes of event B.