From past experience it is known that an investor 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

Answer

To find the probability that an investor will invest neither in security A nor in security B, we can use the following steps:

### Step 1: Define the probabilities
- Let \( P(A) \) be the probability of investing in security A, which is given as \( P(A) = 0.6 \).
- Let \( P(B) \) be the probability of investing in security B, which is given as \( P(B) = 0.3 \).
- Let \( P(A \cap B) \) be the probability of investing in both A and B, which is given as \( P(A \cap B) = 0.2 \).

### Step 2: Use the formula for the union of two events
The probability of investing in either A or B (or both) is given by the formula:
\[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \]

### Step 3: Substitute the known values into the formula
Now, we can substitute the values we have:
\[ P(A \cup B) = 0.6 + 0.3 - 0.2 \]

### Step 4: Calculate \( P(A \cup B) \)
Now, we perform the calculation:
\[ P(A \cup B) = 0.6 + 0.3 - 0.2 = 0.7 \]

### Step 5: Find the probability of investing neither in A nor B
The probability of not investing in either A or B is the complement of the probability of investing in A or B:
\[ P(\text{neither A nor B}) = 1 - P(A \cup B) \]
Substituting the value we found:
\[ P(\text{neither A nor B}) = 1 - 0.7 = 0.3 \]

### Final Answer
Thus, the probability that an investor will invest neither in A nor in B is \( \boxed{0.3} \).

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