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Impala Electronics all 150 employees were asked about their political affiliations. The Employees were grouped by type of work, as Executive or production workers. The results with row and column total shown in the table below. Suppose an employee is selected at random from 150 impala employees. Let us use the following notation for to represent different events of choosing.

E = executive, PW = Production Worker, M = Movement for Multiparty Democracy (MMD) , U = for United Party for National Development (UPND), I = Independent (I)
Below is the employee type and affiliation
Employee Type MMD UPND Independent Row Total
Executive (E) 10 35 10 55
Production Worker 65 22 8 95
Column Total 75 57 18 150

a. Compute P(M) and P(E)
b. Compute P(M/E) i.e probability of M given E
c. Are the events M and E independent? Explain in detail
d. Compute P(M and E)
e. Compute P(M or E)

1 answer

To solve the problem, we first need to analyze the given data from the table and perform the necessary calculations step-by-step.

Given Data:

  • Total Employees: 150
  • Employee Type and Political Affiliations:

| Employee Type | MMD | UPND | Independent | Row Total | |---------------------|-----|------|-------------|-----------| | Executive (E) | 10 | 35 | 10 | 55 | | Production Worker (PW) | 65 | 22 | 8 | 95 | | Column Total | 75 | 57 | 18 | 150 |

a. Compute P(M) and P(E)

To compute \( P(M) \) (the probability an employee is affiliated with MMD) and \( P(E) \) (the probability an employee is an Executive):

  • \( P(M) \): \[ P(M) = \frac{\text{Number of employees affiliated with MMD}}{\text{Total employees}} = \frac{75}{150} = 0.5 \]

  • \( P(E) \): \[ P(E) = \frac{\text{Number of Executive employees}}{\text{Total employees}} = \frac{55}{150} \approx 0.3667 \]

b. Compute P(M | E)

To compute \( P(M | E) \) (the probability of being affiliated with MMD given the employee is an Executive):

\[ P(M | E) = \frac{P(M \text{ and } E)}{P(E)} \]

First, we need to find \( P(M \text{ and } E) \), which is the fraction of employees who are both Executives and affiliated with MMD.

  • Number of Executives affiliated with MMD = 10

\[ P(M \text{ and } E) = \frac{10}{150} = \frac{1}{15} \approx 0.0667 \]

Now, substitute back to find \( P(M | E) \):

\[ P(M | E) = \frac{P(M \text{ and } E)}{P(E)} = \frac{\frac{10}{150}}{\frac{55}{150}} = \frac{10}{55} \approx 0.1818 \]

c. Are the events M and E independent? Explain in detail

Two events A and B are independent if: \[ P(A \text{ and } B) = P(A) \times P(B) \]

Let's check if this holds for M and E:

We previously calculated:

  • \( P(M) = 0.5 \)
  • \( P(E) \approx 0.3667 \)
  • \( P(M \text{ and } E) \approx 0.0667 \)

Now, calculate: \[ P(M) \times P(E) \approx 0.5 \times 0.3667 \approx 0.18335 \]

Since: \[ P(M \text{ and } E) \approx 0.0667 \quad \text{and} \quad P(M) \times P(E) \approx 0.18335 \]

Since \( P(M \text{ and } E) \neq P(M) \times P(E) \), the events M and E are not independent.

d. Compute P(M and E)

We have already calculated \( P(M \text{ and } E) = \frac{10}{150} \) which simplifies to: \[ P(M \text{ and } E) = 0.0667 \quad \text{or} \quad \frac{1}{15}. \]

e. Compute P(M or E)

To compute \( P(M \text{ or } E) \): \[ P(M \text{ or } E) = P(M) + P(E) - P(M \text{ and } E) \]

Substituting the values: \[ P(M \text{ or } E) = 0.5 + 0.3667 - 0.0667 = 0.8 \]

Summary of Results:

  1. \( P(M) = 0.5 \)
  2. \( P(E) \approx 0.3667 \)
  3. \( P(M | E) \approx 0.1818 \)
  4. Events M and E are not independent.
  5. \( P(M \text{ and } E) = 0.0667 \)
  6. \( P(M \text{ or } E) = 0.8 \)