Geometry

Question Analysis
This question involves analyzing a tree diagram to determine the validity of statements regarding probabilities and event relationships in a science - experiment context.
The main focus is on calculating the probabilities of different outcomes, understanding the concept of independent events, and verifying the property of probability sums.

Key Concept Explanation
Probability calculation in a tree diagram: Multiply the probabilities along the branches to find the probability of a particular outcome.
Independent events: Two events are independent if the occurrence of one does not affect the probability of the other.
Sum of probabilities: The sum of probabilities of all possible outcomes in a probability distribution must equal 1.

Step - by - Step Solution
Calculate the probability of getting a positive result:
The probability of a successful setup is 3/4.
Given a successful setup, the probability of a positive result is 2/3.
Multiply these probabilities: (3/4) × (2/3) = 1/2.
Analyze each statement:
Option A: As calculated above, the probability of getting a positive result is 1/2, so this statement is true.
Option B: The probability of an invalid result is the probability of an unsuccessful ...