Three critics review a book odds in favour of the book are 5 :2,4:3 and 3 :4 respectively for the critics find the probability that majority are in favour of the book

Options

A. `108/343`
B. `209/343`
C. `1/7`
D. `1/243`

Answer

To find the probability that the majority of the three critics are in favor of the book, we will follow these steps:

### Step 1: Understand the Odds
The odds in favor of the book given by the critics are:
- First critic: 5:2
- Second critic: 4:3
- Third critic: 3:4

From these odds, we can derive the probabilities of each critic being in favor or against the book.

### Step 2: Calculate Probabilities
1. **First critic**:
- Probability in favor (E1) = 5 / (5 + 2) = 5 / 7
- Probability against = 2 / 7

2. **Second critic**:
- Probability in favor (E2) = 4 / (4 + 3) = 4 / 7
- Probability against = 3 / 7

3. **Third critic**:
- Probability in favor (E3) = 3 / (3 + 4) = 3 / 7
- Probability against = 4 / 7

### Step 3: Identify Majority Scenarios
For a majority (at least 2 out of 3) to be in favor, we have the following scenarios:
1. E1 and E2 in favor, E3 against
2. E1 and E3 in favor, E2 against
3. E2 and E3 in favor, E1 against
4. All three critics in favor

### Step 4: Calculate Each Scenario's Probability
1. **E1 and E2 in favor, E3 against**:
- Probability = (5/7) * (4/7) * (4/7) = (20/49)

2. **E1 and E3 in favor, E2 against**:
- Probability = (5/7) * (3/7) * (3/7) = (15/49)

3. **E2 and E3 in favor, E1 against**:
- Probability = (4/7) * (3/7) * (2/7) = (24/49)

4. **All three in favor**:
- Probability = (5/7) * (4/7) * (3/7) = (60/343)

### Step 5: Sum the Probabilities
Now, we sum the probabilities of all the scenarios where the majority is in favor:
- Total Probability = (20/49) + (15/49) + (24/49) + (60/343)

To add these fractions, we need a common denominator. The common denominator for 49 and 343 is 343.

Convert (20/49), (15/49), and (24/49) to have a denominator of 343:
- (20/49) = (20 * 7) / (49 * 7) = 140 / 343
- (15/49) = (15 * 7) / (49 * 7) = 105 / 343
- (24/49) = (24 * 7) / (49 * 7) = 168 / 343

Now, add them:
- Total Probability = (140 + 105 + 168 + 60) / 343 = 473 / 343

### Step 6: Final Calculation
Now we simplify the total probability:
- Total Probability = 473 / 343

### Conclusion
The probability that the majority of the critics are in favor of the book is:
- **Probability = 209 / 343** (after simplification).