Suppose `E_1, E_2 and E_3` be three mutually exclusive events such that `P(E_i)=p_i" for " i=1, 2, 3`.
P(none of `E_1, E_2, E_3`) equals

Options

A. 0
B. 1-`(p_1+p_2+p_3)`
C. `(1-p_1)(1-p_2)(1-P_3)`
D. None of these

Answer

To find the probability that none of the events \( E_1, E_2, \) and \( E_3 \) occur, we can follow these steps:

### Step 1: Understand the Events
Given that \( E_1, E_2, \) and \( E_3 \) are mutually exclusive events, it means that the occurrence of one event excludes the occurrence of the others.

### Step 2: Write the Probability of None of the Events
The probability of none of the events occurring can be expressed as:
\[ P(\text{none of } E_1, E_2, E_3) = P(E_1^c \cap E_2^c \cap E_3^c) \]
Where \( E_i^c \) denotes the complement of event \( E_i \).

### Step 3: Use the Complement Rule
Using the complement rule, we can express the probability of none of the events occurring in terms of the probability of at least one event occurring:
\[ P(E_1^c \cap E_2^c \cap E_3^c) = 1 - P(E_1 \cup E_2 \cup E_3) \]

### Step 4: Calculate \( P(E_1 \cup E_2 \cup E_3) \)
Since the events are mutually exclusive, the probability of their union can be calculated as:
\[ P(E_1 \cup E_2 \cup E_3) = P(E_1) + P(E_2) + P(E_3) = p_1 + p_2 + p_3 \]

### Step 5: Substitute Back
Now, substituting this back into our equation for the probability of none of the events:
\[ P(\text{none of } E_1, E_2, E_3) = 1 - (p_1 + p_2 + p_3) \]

### Final Answer
Thus, the probability that none of the events \( E_1, E_2, \) and \( E_3 \) occurs is:
\[ P(\text{none of } E_1, E_2, E_3) = 1 - (p_1 + p_2 + p_3) \]

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