
If A and B are two independent events such that
P
(
A
c
)
=
0.67
P(A 
c
 )=0.67, 
P
(
B
c
)
=
x
P(B 
c
 )=x and 
P
(
A
∪
B
)
=
0.9
P(A∪B)=0.9, then calculate the value of 
x
x. (Enter the answer correct to two decimal places)


Given the probabilities for the complements of events \( A \) and \( B \), and the probability of the union of \( A \) and \( B \), we need to find the value of \( x \). Here are the given probabilities:

\[ P(A^c) = 0.67 \]
\[ P(B^c) = x \]
\[ P(A \cup B) = 0.9 \]

First, let's express these in terms of the probabilities of the events themselves. The complement rule tells us:

\[ P(A) = 1 - P(A^c) = 1 - 0.67 = 0.33 \]
\[ P(B) = 1 - P(B^c) = 1 - x \]

Since \( A \) and \( B \) are independent events, the probability of their intersection is:

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

Using the formula for the probability of the union of two events:

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

Substitute the known values into this formula:

\[ 0.9 = P(A) + P(B) - P(A) \cdot P(B) \]

Substitute \( P(A) = 0.33 \) and \( P(B) = 1 - x \):

\[ 0.9 = 0.33 + (1 - x) - (0.33 \cdot (1 - x)) \]

Simplify inside the parentheses:

\[ 0.9 = 0.33 + 1 - x - 0.33 + 0.33x \]

Combine like terms:

\[ 0.9 = 1 - x + 0.33x \]

\[ 0.9 = 1 - x + 0.33x \]

Combine the \( x \) terms:

\[ 0.9 = 1 - 0.67x \]

Solve for \( x \):

\[ 0.9 - 1 = -0.67x \]

\[ -0.1 = -0.67x \]

\[ x = \frac{0.1}{0.67} \]

Calculate the value:

\[ x \approx 0.15 \]

Therefore, the value of \( x \) is approximately \( 0.15 \).

If you have any questions or need further details, please let me know. Here are some related questions:

1. How is the complement rule used in probability?
2. What is the significance of events being independent in probability calculations?
3. Can you explain how the formula for the union of two events is derived?
4. What are the implications if \( A \) and \( B \) are not independent?
5. How would the calculation change if \( P(A \cup B) \) were different?
6. Can you provide an example of dependent events and their probability calculations?
7. How is the intersection of two events related to their individual probabilities?
8. What other properties of probability distributions are useful in solving similar problems?

**Tip:** When dealing with probabilities of complements, always remember that \( P(A^c) = 1 - P(A) \). This is a fundamental concept that simplifies many probability calculations.

Learn how to calculate the probability of independent events with given complements and the union of events. Solve for the complement probability and find the value of x using detailed step-by-step calculations.

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