Asked by df
Marco has a bag of red, blue, and green tiles. Which set of events would be considered independent?
A tile is drawn and replaced, and then a second tile is drawn.
A tile is drawn and removed, and then a second tile is drawn.
A red or blue or green tile is drawn.
Two tiles are drawn at the same time.
All Answers 1
Answered by
GPT-5 mini
AI
"A tile is drawn and replaced, and then a second tile is drawn."
With replacement the composition of the bag is the same for both draws, so the outcome of the first draw does not change the probabilities for the second. Formally, if N is the total number of tiles and r the number of red tiles,
P(first = red) = r/N,
P(second = red) = r/N,
P(both = red) = (r/N)(r/N) = P(first)·P(second),
so P(second | first) = P(second).
By contrast, removing a tile (no replacement) or drawing two at the same time changes the remaining composition and so the draws are generally dependent. (The event "red or blue or green" is the certain event P=1; mathematically it is trivially independent of any other event but is not the intended useful example.)
With replacement the composition of the bag is the same for both draws, so the outcome of the first draw does not change the probabilities for the second. Formally, if N is the total number of tiles and r the number of red tiles,
P(first = red) = r/N,
P(second = red) = r/N,
P(both = red) = (r/N)(r/N) = P(first)·P(second),
so P(second | first) = P(second).
By contrast, removing a tile (no replacement) or drawing two at the same time changes the remaining composition and so the draws are generally dependent. (The event "red or blue or green" is the certain event P=1; mathematically it is trivially independent of any other event but is not the intended useful example.)
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.