Two independent events A and B are given such that P(A)=k, P(B)=k+0.3 and P(A and B) =0.18find k and find P(A or B)

We are given two independent events \( A \) and \( B \) with the following probabilities:

- \( P(A) = k \)
- \( P(B) = k + 0.3 \)
- \( P(A \text{ and } B) = 0.18 \)

For independent events, the probability of \( A \text{ and } B \) is given by the product of their individual probabilities:

\[
P(A \text{ and } B) = P(A) \times P(B)
\]

Substitute the given probabilities:

\[
0.18 = k \times (k + 0.3)
\]

Expanding the equation:

\[
0.18 = k^2 + 0.3k
\]

This is a quadratic equation:

\[
k^2 + 0.3k - 0.18 = 0
\]

Now, we will solve this quadratic equation using the quadratic formula:

\[
k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Where:
- \( a = 1 \)
- \( b = 0.3 \)
- \( c = -0.18 \)

Let's compute the value of \( k \).The solutions for \( k \) are:

\[
k_1 = 0.3 \quad \text{and} \quad k_2 = -0.6
\]

Since probabilities cannot be negative, we take \( k_1 = 0.3 \).

Now, let's calculate \( P(A \text{ or } B) \). The formula for the union of two independent events is:

\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\]

Substitute the known values:

\[
P(A \text{ or } B) = k + (k + 0.3) - 0.18
\]

Substitute \( k = 0.3 \):

\[
P(A \text{ or } B) = 0.3 + (0.3 + 0.3) - 0.18
\]

Let me compute that.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can calculate \( P(A \text{ or } B) \) as follows:

\[
P(A \text{ or } B) = 0.3 + (0.3 + 0.3) - 0.18 = 0.72 - 0.18 = 0.54
\]

Thus, \( P(A \text{ or } B) = 0.54 \).

Let me know if you want further clarification or details!

Here are 5 questions related to this:

1. How do you interpret the independence of two events?
2. What happens if events \( A \) and \( B \) are not independent?
3. How would the solution change if \( P(A \text{ and } B) \) were not given?
4. Can probabilities be greater than 1 or negative?
5. How can the inclusion-exclusion principle help in solving more complex problems?

**Tip:** Always check the conditions for independence when working with probabilities, as this assumption simplifies many calculations.

Two independent events A and B are given such that P(A)=k, P(B)=k+0.3 and P(A and B) =0.18. Find k and find P(A or B).

Learn how to solve a probability problem involving independent events A and B, given P(A)=k, P(B)=k+0.3, and P(A and B)=0.18. Discover how to find the value of k using the quadratic equation and calculate P(A or B) using the inclusion-exclusion principle. Suitable for Grades 9-12.

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