How does drawing a yellow marble change the probability of another yellow marble?

Instruction: A bag contains seven blue and five yellow marbles. Find the initial yellow probability, the second-yellow probability given a yellow first draw without replacement, and the probability of two yellow draws.

Context: Distinguishes initial, conditional and joint probabilities in sampling without replacement.

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Official answer

Initially there are 12 marbles, five yellow, so a uniform first draw is yellow with probability 5/12.

Given that the first marble is yellow and is not replaced, four yellow marbles remain among 11. Thus P(second yellow | first yellow) = 4/11.

The chance that both draws are yellow is (5/12)(4/11) = 5/33. Do not confuse this joint probability with the conditional 4/11. With replacement and a fresh independent uniform draw, the conditional probability would stay 5/12.

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