Let's answer each part step-by-step, using the joint frequency distribution table: | Age Group | Approve | Disapprove | Indifferent | Total | |-------------|---------|------------|-------------|-------| | under 25 | 28 | 30 | 29 | 87 | | 25 to 40 | 33 | 30 | 10 | 73 | | over 40 | 34 | 37 | 30 | 101 | | **Total** | 95 | 97 | 69 | 261 | --- ## 1. What is the probability that a randomly selected employee: ### a) **Did not approve?** - Total employees: 261 - Number who did **not** approve = Total - Approve = 261 - 95 = **166** \[ P(\text{Did not approve}) = \frac{166}{261} \approx 0.6360 \] --- ### b) **Was in the 25 to 40 age group and was indifferent?** - Number in 25 to 40 & Indifferent = 10 \[ P(\text{25 to 40 and Indifferent}) = \frac{10}{261} \approx 0.0383 \] --- ### c) **Was not in the 25 to 40 age group and was indifferent?** - Not in 25 to 40 = (under 25 or over 40) - Indifferent: under 25 = 29, over 40 = 30 → total = 29 + 30 = **59** \[ P(\text{Not 25 to 40 and Indifferent}) = \frac{59}{261} \approx 0.2268 \] --- ### d) **Was in the under 25 age group or was indifferent?** We use the inclusion-exclusion principle: - Under 25: 87 - Indifferent: 69 - Under 25 & Indifferent: 29 \[ P(\text{Under 25 or Indifferent}) = \frac{87 + 69 - 29}{261} = \frac{127}{261} \approx 0.4866 \] --- ### e) **Was not in the under 25 age group or was indifferent?** Again, use inclusion-exclusion: - Not under 25: 261 - 87 = 174 - Indifferent: 69 - Not under 25 & Indifferent: (25-40 & Indif.) + (over 40 & Indif.) = 10 + 30 = 40 \[ P(\text{Not under 25 or Indifferent}) = \frac{174 + 69 - 40}{261} = \frac{203}{261} \approx 0.7778 \] --- ### f) **Was not under 25 and did not approve?** "Not under 25" means 25 to 40 or over 40: - 25 to 40, Disapprove: 30 - 25 to 40, Indifferent: 10 - over 40, Disapprove: 37 - over 40, Indifferent: 30 Add these: 30 + 10 + 37 + 30 = **107** \[ P(\text{Not under 25 and Did not approve}) = \frac{107}{261} \approx 0.4096 \] --- ## 2. **Given that an employee was in the over 40 age group, what is the probability that the employee:** Total in over 40: 101 ### a) **Approved?** - Over 40 & Approved: 34 \[ P(\text{Approved} \mid \text{Over 40}) = \frac{34}{101} \approx 0.3366 \] --- ### b) **Was not indifferent?** - Over 40 & Indifferent: 30 - Not indifferent: 101 - 30 = 71 \[ P(\text{Not Indifferent} \mid \text{Over 40}) = \frac{71}{101} \approx 0.7020 \] --- ## 3. **What is the probability of selecting an employee that was in the under 25 age group from a group of employees that:** ### a) **Was indifferent?** - Indifferent total: 69 - Indifferent & under 25: 29 \[ P(\text{Under 25} \mid \text{Indifferent}) = \frac{29}{69} \approx 0.4203 \] --- ### b) **Did not disapprove?** "Did not disapprove" means Approve or Indifferent. - Number who did not disapprove: Approve (95) + Indifferent (69) = 164 - Under 25 & Approve: 28 - Under 25 & Indifferent: 29 - Total under 25 and did not disapprove: 28 + 29 = 57 \[ P(\text{Under 25} \mid \text{Did not disapprove}) = \frac{57}{164} \approx 0.3476 \] --- # **Summary Table of Answers** | Question | Probability | Calculation | Decimal (4+ places) | |----------|-------------|-------------|---------------------| | 1a | \( \frac{166}{261} \) | (261-95)/261 | **0.6360** | | 1b | \( \frac{10}{261} \) | 10/261 | **0.0383** | | 1c | \( \frac{59}{261} \) | (29+30)/261 | **0.2268** | | 1d | \( \frac{127}{261} \) | (87+69-29)/261 | **0.4866** | | 1e | \( \frac{203}{261} \) | (174+69-40)/261 | **0.7778** | | 1f | \( \frac{107}{261} \) | (30+10+37+30)/261 | **0.4096** | | 2a | \( \frac{34}{101} \) | 34/101 | **0.3366** | | 2b | \( \frac{71}{101} \) | (34+37)/101 | **0.7020** | | 3a | \( \frac{29}{69} \) | 29/69 | **0.4203** | | 3b | \( \frac{57}{164} \) | (28+29)/164 | **0.3476** | If you need any step explained in more detail, let me know!

Certainly! Here's the detailed breakdown divided into three parts, each with explanations: --- ### **Part 1: Calculating Probabilities for General Responses (Questions 1a to 1f)** **Question 1a:** *Probability that a randomly selected employee did not approve* - Total employees: 261 - Number who did not approve: Total - Approved = 261 - 95 = **166** \[ P(\text{Did not approve}) = \frac{166}{261} \approx 0.6360 \] **Question 1b:** *Probability that an employee was in the 25 to 40 age group and was indifferent* - Number in 25-40 & Indifferent: 10 \[ P(\text{25-40 and Indifferent}) = \frac{10}{261} \approx 0.0383 \] **Question 1c:** *Probability that an employee was not in the 25 to 40 age group and was indifferent* - Not in 25-40: (under 25 + over 40) = 87 + 101 = 188 - Indifferent in these groups: under 25 (29), over 40 (30) → total = 59 \[ P(\text{Not 25-40 and Indifferent}) = \frac{59}{261} \approx 0.2268 \] **Question 1d:** *Probability that an employee was in the under 25 age group or was indifferent* - Under 25: 87 - Indifferent: 69 - Both under 25 and indifferent: 29 Using inclusion-exclusion: \[ P(\text{Under 25 or Indifferent}) = \frac{87 + 69 - 29}{261} = \frac{127}{261} \approx 0.4866 \] **Question 1e:** *Probability that an employee was not in under 25 or was indifferent* - Not under 25: 174 (since total 261 - 87) - Indifferent: 69 - Not under 25 & Indifferent: 29 (under 25 & indifferent) + 30 (over 40 & indifferent) = 59 Using inclusion-exclusion: \[ P(\text{Not under 25 or Indifferent}) = \frac{174 + 69 - 59}{261} = \frac{184}{261} \approx 0.7046 \] *(Note: Earlier, we calculated 203/261; rechecking the logic here suggests the total of "not under 25" and "indifferent" minus their intersection = 174 + 69 - 59 = 184. So the probability is 184/261 ≈ 0.7046. Apologies for the earlier miscalculation; the correct calculation uses the intersection of "not under 25" and "indifferent" as 59.)* **Question 1f:** *Probability that an employee was not under 25 and did not approve* - Not under 25: 174 - Did not approve: 166 - Employees who are not under 25 and did not approve: sum of disapprove in 25-40 and over 40 groups: - 25-40 & Disapprove: 30 - Over 40 & Disapprove: 37 - 25-40 & Indifferent, Over 40 & Indifferent: not counted here - But since "did not approve" includes disapprove and indifferent, total "not approve" in these groups is 30 + 37 + (over 40 & Indifferent: 30) but we need only those who are "not under 25" and "did not approve" (disapprove or indifferent). Actually, "did not approve" = disapprove + indifferent = 97 total. Total employees who are "not under 25" and "did not approve" = disapprove + indifferent in 25-40 and over 40: - 25-40 & Disapprove: 30 - 25-40 & Indifferent: 10 - over 40 & Disapprove: 37 - over 40 & Indifferent: 30 Sum: 30+10+37+30=107 \[ P(\text{Not under 25 and Did not approve}) = \frac{107}{261} \approx 0.4096 \] **_Explanation:_** In this part, we focused on basic probability calculations. We identified relevant counts from the table, applied the total sample size, and used inclusion-exclusion where necessary. The key is understanding how to combine counts to find joint and marginal probabilities. --- ### **Part 2: Conditional Probabilities

Certainly! Here's the continuation with detailed explanations for the remaining parts: --- ### **Part 2: Conditional Probabilities Given the Over 40 Age Group (Questions 2a and 2b)** **Question 2a:** *Given that an employee was in the over 40 age group, what is the probability they approved?* - Total over 40: **101** - Number of over 40 employees who approved: **34** \[ P(\text{Approved} \mid \text{Over 40}) = \frac{34}{101} \approx 0.3366 \] **_Explanation:_** Conditional probability is calculated as the ratio of the number of favorable outcomes (over 40 & approved) to the total number of outcomes in the given condition (over 40). This reflects the likelihood that an employee in the over 40 group approved, based solely on the subgroup. --- **Question 2b:** *Given that an employee was in the over 40 age group, what is the probability they were not indifferent?* - Total over 40: 101 - Over 40 & Indifferent: 30 - Over 40 & Not Indifferent: 101 - 30 = 71 \[ P(\text{Not Indifferent} \mid \text{Over 40}) = \frac{71}{101} \approx 0.7020 \] **_Explanation:_** Here, we identify the number of over 40 employees who are **not indifferent** (either approve or disapprove). Dividing by the total in the over 40 group gives the probability that a randomly chosen over 40 employee was not indifferent, reflecting their likelihood of having a definite opinion. --- ### **Part 3: Conditional Probability of Under 25 Given Specific Response Types (Questions 3a and 3b)** **Question 3a:** *From employees who were indifferent, what is the probability that the employee was in the under 25 age group?* - Total indifferent: 69 - Under 25 & Indifferent: 29 \[ P(\text{Under 25} \mid \text{Indifferent}) = \frac{29}{69} \approx 0.4203 \] **_Explanation:_** This measures the likelihood that an indifferently responding employee is from the under 25 age group. It’s a simple conditional probability where we focus only on the indifferent subgroup and see how many are under 25. --- **Question 3b:** *From employees who did not disapprove (approved or indifferent), what is the probability that the employee was in the under 25 age group?* - Total not disapprove: Approve + Indifferent = 95 + 69 = 164 - Under 25 & Approve: 28 - Under 25 & Indifferent: 29 - Total under 25 & not disapprove: 28 + 29 = 57 \[ P(\text{Under 25} \mid \text{Not disapprove}) = \frac{57}{164} \approx 0.3476 \] **_Explanation:_** This probability tells us, among employees who did not disapprove (either approved or indifferent), how likely it is that they are from the under 25 group. It's useful for understanding the composition of supportive or neutral responses within that subgroup. --- ### **Summary of Part 2 and 3:** | Question | Probability | Explanation | |------------|--------------|-------------| | 2a | \( \frac{34}{101} \approx 0.3366 \) | Likelihood an over 40 employee approved | | 2b | \( \frac{71}{101} \approx 0.7020 \) | Likelihood an over 40 employee was not indifferent | | 3a | \( \frac{29}{69} \approx 0.4203 \) | Likelihood an indifferent employee is under 25 | | 3b | \( \frac{57}{164} \approx 0.3476 \) | Likelihood an employee not disapproving is under 25 | --- If you'd like, I can prepare a final wrap-up or further explanations!