If A and B are two independent events such that `P(A cup B) = 0.8 and P(A) = 0.3` then P(B) is equal to: a) 2/3 b) 5/7 c) 1/8 d) 3/8

Options

A. `2/3`
B. `5/7`
C. `1/8`
D. `3/8`

Answer

To find \( P(B) \) given that \( P(A \cup B) = 0.8 \) and \( P(A) = 0.3 \), we can use the formula for the probability of the union of two independent events:

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

Since \( A \) and \( B \) are independent events, we have:

\[ P(A \cap B) = P(A) \times P(B) \]

Substituting this into the union formula, we get:

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

Now, substituting the known values:

\[ 0.8 = 0.3 + P(B) - 0.3 \times P(B) \]

Let's simplify this equation step by step:

1. Combine like terms:
\[ 0.8 = 0.3 + P(B) - 0.3P(B) \]
\[ 0.8 = 0.3 + P(B)(1 - 0.3) \]
\[ 0.8 = 0.3 + 0.7P(B) \]

2. Isolate \( P(B) \):
\[ 0.8 - 0.3 = 0.7P(B) \]
\[ 0.5 = 0.7P(B) \]

3. Solve for \( P(B) \):
\[ P(B) = \frac{0.5}{0.7} \]
\[ P(B) = \frac{5}{7} \]

Thus, the probability \( P(B) \) is \( \frac{5}{7} \).