Two gold marbles back to back

Question

A bag holds 8 red marbles and 2 gold marbles, 10 in total. You draw 2 marbles one at a time, without replacement. What is the probability that the first draw is gold and the second draw is also gold?

Accepted Answer

Order matters, so multiply the two draws in turn, updating the bag after the first. First draw gold. Only 2 gold out of 10: \frac{2}{10}. Second draw gold. One gold gone, so just 1 gold left out of 9: \frac{1}{9}. Multiply: \frac{2}{10}	imes\frac{1}{9}=\frac{2}{90}=\frac{1}{45}\approx 0.022. Takeaway. When only two of the target exist, the second draw is a razor-thin 1-in-9, so the sequence is rare. Scarcity compounds fast without replacement.

Two gold marbles back to back

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