If A and B are two mutually exclusive events, then the relation between `P(bar A) and P(B)` is

Options

A. `P(A) le P(overline(B))`
B. `P(A) gt P(overlineB)`
C. `P(A) lt P(B)`
D. none of these

Explanation

To solve the problem, we need to establish the relationship between \( P(\bar{A}) \) (the probability of the complement of event A) and \( P(B) \) (the probability of event B) when A and B are mutually exclusive events.

### Step 1: Understand Mutually Exclusive Events
Mutually exclusive events are events that cannot occur at the same time. This means that if one event occurs, the other cannot. Mathematically, this is represented as:
\[ P(A \cap B) = 0 \]

### Step 2: Use the Complement Rule
The probability of the complement of an event A is given by:
\[ P(\bar{A}) = 1 - P(A) \]
This means that \( P(\bar{A}) \) represents the probability of all outcomes that are not in A.

### Step 3: Relationship Between Events A and B
Since A and B are mutually exclusive, we can express the total probability of the sample space as:
\[ P(A) + P(B) + P(\text{other events}) = 1 \]
If we denote the probability of other events as \( P(O) \), we can rewrite it as:
\[ P(A) + P(B) + P(O) = 1 \]

### Step 4: Express \( P(B) \) in Terms of \( P(\bar{A}) \)
From the total probability, we can express \( P(O) \) as:
\[ P(O) = 1 - P(A) - P(B) \]
Now substituting \( P(O) \) back into the equation for \( P(\bar{A}) \):
\[ P(\bar{A}) = 1 - P(A) = P(B) + P(O) \]
This shows that \( P(\bar{A}) \) includes \( P(B) \) and the probabilities of other events.

### Step 5: Conclusion
Since \( P(B) \) is a part of \( P(\bar{A}) \) and considering that \( P(A) + P(B) \) can be less than or equal to 1, we can conclude:
\[ P(\bar{A}) \geq P(B) \]
Thus, the relation between \( P(\bar{A}) \) and \( P(B) \) is:
\[ P(\bar{A}) \geq P(B) \]