Question

If probability distribution is given by \[ P(x) = \begin{array}{c|c|c|c|c|c|c|c} x & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ \hline P(x) & k & 2k^2 & 6k^2 & 2k^2 + k & 4k & k & k \\ \end{array} \] Then, the value of \( P(3<x \leq 6) \) is:

• A. 0.6
• B. 0.8
• C. 0.4
• D. 0.2

Hint:

When dealing with probability distributions, always ensure that the total probability sums to 1, and use this to solve for unknown constants.

Answer 1

The Correct Option is A

Step 1: Find the constant \( k \).
To find the value of \( k \), use the condition that the total probability must sum to 1: \[ \sum_{x=0}^{7} P(x) = 1 \] Solve for \( k \) by substituting the values of \( P(x) \) for each \( x \). Step 2: Calculate \( P(3<x \leq 6) \).
We need to calculate the sum of the probabilities for \( x = 4, 5, 6 \): \[ P(3<x \leq 6) = P(4) + P(5) + P(6) \] Step 3: Conclusion.
Thus, the value of \( P(3<x \leq 6) \) is 0.6. Final Answer: \[ \boxed{0.6} \]