Problem 73 - Basic Statistics for Business and Economics with Student

Question

There are four people being considered for the position of chief executive officer of Dalton Enterprises. Three of the applicants are over 60 years of age. Two are female, of which only one is over 60 . a. What is the probability that a candidate is over 60 and female? b. Given that the candidate is male, what is the probability he is less than \(60 ?\) c. Given that the person is over \(60,\) what is the probability the person is female?

Step-by-Step Solution

Answer

a. \( \frac{1}{4} \), b. \( \frac{1}{2} \), c. \( \frac{1}{3} \)

Step 1: Define the Given Information

Let's begin by defining the information given in the problem. There are 4 candidates. 3 candidates are over 60 years old (we'll call this event A). 2 candidates are female (we'll call this event B), and among the females, only 1 is over 60.

Step 2: Calculate the Total Possibilities

There are 4 candidates in total. We will investigate the distribution of these candidates based on age and gender. Out of 4 candidates, 3 are over 60, so 1 candidate is under 60. Similarly, 2 candidates are females, and thus 2 are males.

Step 3: Calculate the Probability a Candidate is Over 60 and Female (Part a)

Let's find the intersection of the events where the candidate is over 60 and female. From Step 1, we know that only 1 female candidate is over 60. Thus, the probability that a candidate is both over 60 and female is 1 out of 4 candidates, which is \( \frac{1}{4} \).

Step 4: Calculate the Probability a Male Candidate is Less than 60 (Part b)

Since there is only 1 candidate who is less than 60 and 2 male candidates, we now determine if this candidate not over 60 is male. Given that a candidate is male, we know there are 2 males out of 4 candidates. Since 1 candidate is both male and less than 60, the probability of a male being under 60 is \( \frac{1}{2} \).

Step 5: Calculate the Probability of being Female given over 60 (Part c)

Three candidates are over 60, including 1 female, so the probability that a candidate is over 60 and female is \( \frac{1}{3} \) since there is 1 female over 60 out of 3 total candidates over 60.

Key Concepts

Conditional Probability

When attempting to find the likelihood of an event occurring given that another event has already happened, we use conditional probability. In this exercise, we encounter several instances where conditional probability is applicable. Let's take a deeper look at one example to explore this concept fully.

In part (b), we look at the probability a male candidate is less than 60 years old, given that we already know the candidate is male. Here, our sample space shrinks from all 4 candidates to only the 2 male candidates.

• Original sample: 4 candidates.
• Relevant sample: 2 male candidates.

Given this reduced sample space, we identify that only 1 male is under 60.

The probability a male is under 60 is then the number of favorable outcomes (male under 60) divided by the total possible outcomes in our reduced sample (total males), giving us \(\frac{1}{2}\).

Conditional probability is calculated using the formula:\[P(A|B) = \frac{P(A \cap B)}{P(B)}\]
Understanding conditional probability necessitates carefully identifying which conditions limit your sample space and adjusting your calculations accordingly.

Age and Gender Distribution

The exercise gives us significant cues concerning the distribution of both age and gender among the candidates. Breaking down these details can provide a clear picture of how to approach similar problems.

Step 1: Understand the Age Distribution
We know from the exercise:

• 3 candidates are over 60 years old.
• 1 candidate is therefore under 60.

This means age distinctions are not spread evenly among candidates, playing a crucial role in calculating probabilities pertaining to these age groups.

Step 2: Grasp the Gender Distribution
According to the exercise:

• 2 are females.
• Thus, 2 must be males.

The gender split allows us to examine other parameters, such as how many females are over 60, crucial for the calculations.

Step 3: Combining Age and Gender
We learn that only 1 female candidate is over 60. Therefore, the interplay between age and gender assists in isolating specific candidates for probability calculations. This blending of characteristics helps refine our interpretation of data, enabling precise computation.

Basic Statistics

Basic statistics is the cornerstone of interpreting data in problems like this one. At its core, it involves understanding the total and part relationships within a sample set, essential for calculating probabilities.

Consistent Sample Set
Our sample set contains 4 individuals; hence, comparisons and probability assessments are made from this fixed number:

• Counting this accurately is paramount: misstatements can lead to erroneous results.
• Being mindful of the constant sample size aids clarity in probability determination.

Intersection of Events
In part (a), the statistical task is identifying the intersection, where a candidate is both female and over 60. This context sharpens focus on how to approach the calculation:

• With 1 out of 4 candidates fitting this dual criterion, probability becomes \(\frac{1}{4}\).
• This concept underlies the importance of isolating intersecting features properly."

Logical Structure
Formulating a logical structure hinges on delineating events and their subsets meticulously, employing core statistical principles as guides. Ensuring nuances are understood ensures the validity of probabilistic estimations and their breakdown."