Suppose q=0.13. Find P(1

Question

Suppose q=0.13. Find P(1 < X <= 3).

BlackTom AI · Accepted Answer

This item is a short-answer question, and the prompt as given does not include the distribution of X or any information that would let us compute P(1 < X ≤ 3) directly. To reason through it, we first identify what information is required to compute any probability about X: we need the distribution (or at least the cumulative distribution function, CDF) of X, or the probability mass function (PMF) if X is discrete.

Option 1: Determine what information would allow calculation. If X has a known discrete distribution, we could express P(1 < X ≤ 3) as P(X ≤ 3) − P(X ≤ 1) using the CDF, or as the sum P(X = 2) + P(X = 3) if that form is convenient depending on how the distribution is presented.

Option 2: Consider common distributions and see what P(1 < X ≤ 3) would look like for the given parameter q = 0.13. If X were geometric (counting number of trials up to and including the first success) with success probability q, then P(1 < X ≤ 3) would be P(X ≤ 3) − P(X ≤ 1). Depending on the version of the geometric distribution (support starting at 1 or at 0 for the number of failures before the first success), the exact expression changes. For the version where X has support {1, 2, 3, ...}, P(X ≤ k) = 1 − (1 − q)^k, so P(1 < X ≤ 3) = [1 − (1 − q)^3] − [1 − (1 − q)^1] = (1 − q) − (1 − q)^3. Substituting q = 0.13 gives (0.87) − (0.87)^3 ≈ 0.2115. Notice this result does not equal 0.67, suggesting either a different distribution or a misgiven parameter value would be at play.

Option 3: Identify what would happen if no distribution information is available. In that case, you cannot compute P(1 < X ≤ 3) uniquely; the probability depends on how X is distributed. All you could do is state the need for X’s distribution and request it from the problem source.

Option 4: Reflect on the provided answer 0.67. If the problem statement truly intends a calculation using q = 0.13, and if X’s distribution is unspecified here, then achieving 0.67 would require an alternative interpretation or a different distribution than the ones commonly used, or there may be a transcription error in the question or answer. Without clarifying information, treating 0.67 as correct would be unjustified.

In sum, the key steps are: (1) identify or obtain the distribution of X, (2) translate P(1 < X ≤ 3) into a form involving the CDF or PMF, and (3) compute using the given parameter(s). Absent distribution details, any numeric result cannot be guaranteed to be correct, and the stated value 0.67 cannot be verified from q = 0.13 alone.

COMM_V 190 101-104 2025W1 Final Exam - Practice Questions - Second Half

Suppose q=0.13. Find P(1 < X <= 3).

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