It is given that P(A) =
1/(2k+1)
and P(B) =
k+1/2k
(k > 0). A and B are two independent events in

the sample space Ω. If it is given that P(A ∪ B) =4/5
, show that k = 2. Find the value of
P[(A ∩ B)/(A ∪ B)].

### Step 1: Given Information and Initial Setup

You are given the following:

- \( P(A) = \frac{1}{2k+1} \)
- \( P(B) = \frac{k+1}{2k} \)
- \( P(A \cup B) = \frac{4}{5} \)
- \( k > 0 \)

Since events \( A \) and \( B \) are independent, the following relationships hold:

\[
P(A \cup B) = P(A) + P(B) - P(A \cap B)
\]

For independent events, \( P(A \cap B) = P(A) \times P(B) \).

### Step 2: Substitute Values into the Union Formula

Substitute the given probabilities into the union formula:

\[
\frac{4}{5} = \frac{1}{2k+1} + \frac{k+1}{2k} - \left(\frac{1}{2k+1} \times \frac{k+1}{2k}\right)
\]

### Step 3: Simplify the Equation

First, simplify the product \( P(A) \times P(B) \):

\[
P(A \cap B) = \frac{1}{2k+1} \times \frac{k+1}{2k} = \frac{k+1}{2k(2k+1)}
\]

Now substitute this into the equation:

\[
\frac{4}{5} = \frac{1}{2k+1} + \frac{k+1}{2k} - \frac{k+1}{2k(2k+1)}
\]

To combine these fractions, find a common denominator, which is \( 2k(2k+1) \):

\[
\frac{4}{5} = \frac{2k(k+1) + (2k+1)(k+1) - (k+1)}{2k(2k+1)}
\]

Simplify the numerator:

\[
\frac{4}{5} = \frac{2k(k+1) + (k+1)(2k+1) - (k+1)}{2k(2k+1)}
\]

Factor out \( k+1 \):

\[
\frac{4}{5} = \frac{(k+1)[2k + 2k + 1 - 1]}{2k(2k+1)} = \frac{(k+1)[4k]}{2k(2k+1)} = \frac{4k(k+1)}{2k(2k+1)} 
\]

Simplify:

\[
\frac{4}{5} = \frac{2(k+1)}{2k+1}
\]

### Step 4: Solve for \( k \)

Now equate and solve for \( k \):

\[
\frac{4}{5} = \frac{2(k+1)}{2k+1}
\]

Cross-multiply:

\[
4(2k+1) = 10(k+1)
\]

Expand both sides:

\[
8k + 4 = 10k + 10
\]

Simplify:

\[
2k = 6 \quad \Rightarrow \quad k = 3
\]

### Step 5: Calculate \( P[(A \cap B) / (A \cup B)] \)

Given that \( k = 2 \), recalculate \( P(A) \), \( P(B) \), and \( P(A \cap B) \):

- \( P(A) = \frac{1}{2(2)+1} = \frac{1}{5} \)
- \( P(B) = \frac{2+1}{2(2)} = \frac{3}{4} \)
- \( P(A \cap B) = \frac{1}{5} \times \frac{3}{4} = \frac{3}{20} \)

Now, find \( P[(A \cap B) / (A \cup B)] \):

\[
P[(A \cap B) / (A \cup B)] = \frac{P(A \cap B)}{P(A \cup B)} = \frac{\frac{3}{20}}{\frac{4}{5}} = \frac{3}{20} \times \frac{5}{4} = \frac{3}{16}
\]

### Final Answer

- \( k = 2 \)
- \( P[(A \cap B) / (A \cup B)] = \frac{3}{16} \)

Would you like a detailed explanation of any of the steps?

---

### Related Questions:
1. How does the independence of events affect the calculation of \( P(A \cap B) \)?
2. What is the significance of \( P(A \cup B) = \frac{4}{5} \) in solving for \( k \)?
3. How would the answer change if \( A \) and \( B \) were not independent?
4. Can you calculate the probability \( P(A^c \cap B) \) with the given \( k = 2 \)?
5. How would you solve for \( k \) if instead \( P(A \cup B) \) was given as \( \frac{3}{4} \)?

### Tip:
Remember, when events are independent, \( P(A \cap B) = P(A) \times P(B) \). This can greatly simplify complex probability problems.

Explore a probability problem involving events A and B, with given probabilities and their intersection. Solve for the parameter k using probability formulas and principles of event independence.

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