Mathematics
Mathematics
Problem
A box contains 3 red balls and 2 blue balls. Two draws are made in succession from the box. a. (i) State the definition of independent events in probability theory. b. (ii) If the first ball is not replaced, calculate the probability that both the first and the second ball drawn are red. c. (iii) For the situation in part (ii), find the probability that the second ball is red given that the first ball was red. d. (iv) Using your results in parts (ii) and (iii), determine whether the events “first ball is red” and “second ball is red” are independent when the first ball is not replaced. Justify your answer. e. (v) Now suppose the first ball is replaced before the second draw. Repeat the calculations of parts (ii) and (iii) for this new situation and comment on the independence of the two events in this case.Intuitive explanation
The key here is recognizing that conditional probability changes everything: once you draw the first ball, the box composition shifts, so the second draw depends on what happened first. You'll need to track all possible sequences (red-then-red, red-then-blue, etc.) and their individual probabilities, then combine them based on what the problem asks. Think of it as a branching tree where each path has its own probability.
Intuitive explanation
The key here is recognizing that conditional probability changes everything: once you draw the first ball, the box composition shifts, so the second draw depends on what happened first. You'll need to track all possible sequences (red-then-red, red-then-blue, etc.) and their individual probabilities, then combine them based on what the problem asks. Think of it as a branching tree where each path has its own probability.