For A,B and C, the chances of being selected as the manager of a firm are `4:1:2` respectively. The probabilites for them to introduce a radical change in the marketing strategy are `0.3,0.8` and `0.5` respectively. If a change takes place, find the probability that it is due to the appoinment of B.

Answer

To solve the problem, we will use Bayes' theorem. Let's break down the steps:

### Step 1: Define the Events
Let:
- \( E_1 \): Event that A is selected as the manager.
- \( E_2 \): Event that B is selected as the manager.
- \( E_3 \): Event that C is selected as the manager.
- \( E \): Event that a radical change occurs in the marketing strategy.

### Step 2: Calculate the Probabilities of Selection
The chances of being selected as the manager are given in the ratio \( 4:1:2 \). Thus, we can find the probabilities:
- Total parts = \( 4 + 1 + 2 = 7 \)

So,
- \( P(E_1) = \frac{4}{7} \)
- \( P(E_2) = \frac{1}{7} \)
- \( P(E_3) = \frac{2}{7} \)

### Step 3: Probabilities of Introducing Change
The probabilities for A, B, and C to introduce a radical change in the marketing strategy are given as:
- \( P(E | E_1) = 0.3 \)
- \( P(E | E_2) = 0.8 \)
- \( P(E | E_3) = 0.5 \)

### Step 4: Apply Bayes' Theorem
We need to find the probability that the change is due to the appointment of B, given that a change has occurred, i.e., \( P(E_2 | E) \).

Using Bayes' theorem:
\[ P(E_2 | E) = \frac{P(E | E_2) \cdot P(E_2)}{P(E)} \]

### Step 5: Calculate \( P(E) \)
We need to calculate \( P(E) \) using the law of total probability:
\[ P(E) = P(E | E_1) \cdot P(E_1) + P(E | E_2) \cdot P(E_2) + P(E | E_3) \cdot P(E_3) \]

Substituting the values:
\[ P(E) = (0.3 \cdot \frac{4}{7}) + (0.8 \cdot \frac{1}{7}) + (0.5 \cdot \frac{2}{7}) \]
\[ = \frac{1.2}{7} + \frac{0.8}{7} + \frac{1.0}{7} = \frac{3.0}{7} \]

### Step 6: Substitute Values into Bayes' Theorem
Now substitute back into Bayes' theorem:
\[ P(E_2 | E) = \frac{P(E | E_2) \cdot P(E_2)}{P(E)} = \frac{0.8 \cdot \frac{1}{7}}{\frac{3.0}{7}} \]
\[ = \frac{0.8}{3.0} = \frac{8}{30} = \frac{4}{15} \]

### Final Answer
Thus, the probability that the change is due to the appointment of B is:
\[ \boxed{\frac{4}{15}} \]