Two sets of candidates are competing for the position on the board of directors of a company. The probabilities that the first and second sets will win are 0.6 and 0.4 respectively. If the first set wins, the probability of introducing a new product is 0.8 and the corresponding probability if the second set wins is 0.3. What is the probability that the new product will be introduced ?

Options

A. 0.8
B. 0.7
C. 0.6
D. 0.3

Answer

To solve the problem, we need to find the probability that a new product will be introduced based on the winning probabilities of two sets of candidates and their respective probabilities of introducing the product.

### Step-by-Step Solution:

1. **Identify the Given Probabilities:**
- Probability that the first set wins, \( P(A) = 0.6 \)
- Probability that the second set wins, \( P(B) = 0.4 \)
- Probability of introducing a new product if the first set wins, \( P(P|A) = 0.8 \)
- Probability of introducing a new product if the second set wins, \( P(P|B) = 0.3 \)

2. **Use the Law of Total Probability:**
The total probability of the new product being introduced can be calculated using the law of total probability:
\[ P(P) = P(P|A) \cdot P(A) + P(P|B) \cdot P(B) \]

3. **Substitute the Values:**
Substitute the known probabilities into the equation:
\[ P(P) = (0.8 \cdot 0.6) + (0.3 \cdot 0.4) \]

4. **Calculate Each Term:**
- Calculate \( 0.8 \cdot 0.6 \):
\[ 0.8 \cdot 0.6 = 0.48 \]
- Calculate \( 0.3 \cdot 0.4 \):
\[ 0.3 \cdot 0.4 = 0.12 \]

5. **Add the Results:**
Now, add the two results together:
\[ P(P) = 0.48 + 0.12 = 0.60 \]

6. **Final Result:**
Therefore, the probability that the new product will be introduced is:
\[ P(P) = 0.60 \]

### Final Answer:
The probability that the new product will be introduced is **0.60**.