From past experience it is known that an invester will invest in security A with a probability of 0.6, will invest in security B with a probability 0.3 and will invest in both A and B with a probability of 0.2. What is the probability that an investor will invest neither in A nor in B ?

Options

A. 0.7
B. 0.28
C. 0.3
D. 0.4

Explanation

To solve the problem step by step, we need to find the probability that an investor will invest neither in security A nor in security B. We are given the following probabilities:

- Probability of investing in A, P(A) = 0.6
- Probability of investing in B, P(B) = 0.3
- Probability of investing in both A and B, P(A ∩ B) = 0.2

### Step 1: Calculate the Probability of A Union B

We need to find the probability of investing in either A or B, which is represented as P(A ∪ B). We can use the formula for the probability of the union of two events:

\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

Substituting the given values:

\[ P(A \cup B) = 0.6 + 0.3 - 0.2 \]

### Step 2: Perform the Calculation

Now, we calculate:

\[ P(A \cup B) = 0.6 + 0.3 - 0.2 = 0.7 \]

### Step 3: Calculate the Probability of Neither A nor B

To find the probability that the investor will invest neither in A nor in B, we can use the complement rule:

\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \]

Substituting the value we found for P(A ∪ B):

\[ P(\text{neither A nor B}) = 1 - 0.7 \]

### Step 4: Perform the Final Calculation

Now, we calculate:

\[ P(\text{neither A nor B}) = 1 - 0.7 = 0.3 \]

### Conclusion

The probability that an investor will invest neither in A nor in B is **0.3**.

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