The events `E _(1) and E _(2)` have probabilities `0.25 and 0. 50` respectively. If the probability that both `E _(1) and E _(2)` occur simultaneous is `0. 14.` Then the probability that neither `E _(1)` nor `E _(2)` occurs is

Options

A. `0. 39`
B. `0.25`
C. `0.11`
D. None of these

Answer

To find the probability that neither event \( E_1 \) nor event \( E_2 \) occurs, we can use the formula derived from De Morgan's laws. Here’s a step-by-step solution:

### Step 1: Identify the given probabilities
- Probability of event \( E_1 \): \( P(E_1) = 0.25 \)
- Probability of event \( E_2 \): \( P(E_2) = 0.50 \)
- Probability of both events occurring simultaneously: \( P(E_1 \cap E_2) = 0.14 \)

### Step 2: Use the formula for the union of two events
We need to find the probability of at least one of the events occurring, which is given by:
\[ P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2) \]

### Step 3: Substitute the values into the formula
Substituting the known values:
\[ P(E_1 \cup E_2) = 0.25 + 0.50 - 0.14 \]

### Step 4: Calculate the probability of the union
Calculating the right-hand side:
\[ P(E_1 \cup E_2) = 0.75 - 0.14 = 0.61 \]

### Step 5: Find the probability that neither event occurs
The probability that neither event \( E_1 \) nor event \( E_2 \) occurs is the complement of the probability that at least one of them occurs:
\[ P(E_1' \cap E_2') = 1 - P(E_1 \cup E_2) \]
Substituting the value we found:
\[ P(E_1' \cap E_2') = 1 - 0.61 = 0.39 \]

### Final Answer
The probability that neither \( E_1 \) nor \( E_2 \) occurs is \( \boxed{0.39} \).

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