Let X be a uniform random variable on [0,1]. Determine Pr(X ≤1 / 2 |1 / 4 ≤X ≤3 / 4) and Pr(X ≤1

Name: Problem 3, Chapter 8: Probability and Computing
Uploaded: 2025-11-05T17:51:34-08:00
Duration: 47 s
Channel: Hoan Nguyen
Description: Let X be a uniform random variable on [0,1]. Determine Pr(X ≤1 / 2 |1 / 4 ≤X ≤3 / 4) and Pr(X ≤1 / 4 |(X ≤1 / 3) ∪(X ≥2 / 3)).

step 1 All right solution we have D1 intersect D2 is empty and D1 Union D2 so X X and X X and 1 which i

Question

Let $X$ be a uniform random variable on $[0,1]$. Determine $\operatorname{Pr}(X \leq 1 / 2 \mid$ $1 / 4 \leq X \leq 3 / 4)$ and $\operatorname{Pr}(X \leq 1 / 4 \mid(X \leq 1 / 3) \cup(X \geq 2 / 3))$.

   Let $X$ be a uniform random variable on $[0,1]$. Determine $\operatorname{Pr}(X \leq 1 / 2 \mid$ $1 / 4 \leq X \leq 3 / 4)$ and $\operatorname{Pr}(X \leq 1 / 4 \mid(X \leq 1 / 3) \cup(X \geq 2 / 3))$.

Probability and Computing

Michael Mitzenmacher… 2005 Edition

Chapter 8, Problem 3

Instant Answer

Solved by Expert Hoan Nguyen

11/05/2025

Step 1

This is a conditional probability, and we can use the definition of conditional probability: \[ \operatorname{Pr}(A \mid B) = \frac{\operatorname{Pr}(A \cap B)}{\operatorname{Pr}(B)} \] where $A$ is the event $X \leq 1/2$ and $B$ is the event \(1/4 \leq X Show more…

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Let $X$ be a uniform random variable on $[0,1]$. Determine $\operatorname{Pr}(X \leq 1 / 2 \mid$ $1 / 4 \leq X \leq 3 / 4)$ and $\operatorname{Pr}(X \leq 1 / 4 \mid(X \leq 1 / 3) \cup(X \geq 2 / 3))$.

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Key Concepts

Uniform Distribution

A uniform distribution on an interval means that every subinterval of equal length has equal probability. For a continuous uniform random variable over [0, 1], the probability of the variable falling in any subinterval is directly proportional to the length of that subinterval.

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is calculated as the probability of both events happening divided by the probability of the given event, which simplifies probability problems by narrowing the sample space.

Measure Through Interval Lengths

In the context of a uniform distribution over an interval, computing the probability of events reduces to determining the lengths of the intervals involved. This approach takes advantage of the constant probability density, making it easy to handle intersections and unions of events by working with their lengths.

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