If A and B are two independent events such that p(a) = 1/6 and P(b) = 1/2, then

Options

A. P(AB) = 1/24
B. `P(AcupB)=1//12`
C. `P(A^(C)B)=1//6`
D. `P(A^(C)B^(C))=5//12`

Answer

To solve the problem step by step, we will analyze the given probabilities of independent events A and B, and calculate the required probabilities.

### Step 1: Identify the given probabilities
We are given:
- \( P(A) = \frac{1}{6} \)
- \( P(B) = \frac{1}{2} \)

### Step 2: Calculate the probabilities of the complements
The complement of an event A, denoted as \( A' \) or \( A^c \), is calculated as:
\[ P(A') = 1 - P(A) \]
Thus:
\[ P(A') = 1 - \frac{1}{6} = \frac{5}{6} \]

Similarly, for event B:
\[ P(B') = 1 - P(B) \]
Thus:
\[ P(B') = 1 - \frac{1}{2} = \frac{1}{2} \]

### Step 3: Calculate the probability of both A and B occurring (AB)
Since A and B are independent events, the probability of both A and B occurring is given by:
\[ P(AB) = P(A) \times P(B) \]
Calculating this:
\[ P(AB) = \frac{1}{6} \times \frac{1}{2} = \frac{1}{12} \]

### Step 4: Calculate the probability of A union B (A ∪ B)
The probability of either A or B occurring is given by:
\[ P(A \cup B) = P(A) + P(B) - P(AB) \]
Substituting the values we have:
\[ P(A \cup B) = \frac{1}{6} + \frac{1}{2} - \frac{1}{12} \]
To add these fractions, we need a common denominator, which is 12:
\[ P(A \cup B) = \frac{2}{12} + \frac{6}{12} - \frac{1}{12} = \frac{7}{12} \]

### Step 5: Calculate the probability of A' and B occurring (A'B)
Since A and B are independent, we can write:
\[ P(A'B) = P(A') \times P(B) \]
Calculating this:
\[ P(A'B) = \frac{5}{6} \times \frac{1}{2} = \frac{5}{12} \]

### Step 6: Calculate the probability of A' and B' occurring (A'B')
Again, using independence:
\[ P(A'B') = P(A') \times P(B') \]
Calculating this:
\[ P(A'B') = \frac{5}{6} \times \frac{1}{2} = \frac{5}{12} \]

### Summary of Results
- \( P(AB) = \frac{1}{12} \)
- \( P(A \cup B) = \frac{7}{12} \)
- \( P(A'B) = \frac{5}{12} \)
- \( P(A'B') = \frac{5}{12} \)

### Final Answers
1. \( P(AB) = \frac{1}{12} \) (not matching the given)
2. \( P(A \cup B) = \frac{7}{12} \) (not matching the given)
3. \( P(A'B) = \frac{5}{12} \) (this is correct)
4. \( P(A'B') = \frac{5}{12} \) (this is also correct)