Drawing black, white, white in order

Question

A box holds 7 white balls and 3 black balls, 10 in total. You draw 3 balls one at a time, without putting any back.

What is the probability that the draws come out in the exact order black, white, white?

Accepted Answer

Order matters, so multiply the chances draw by draw, shrinking the box as balls leave. First draw black. 3 black out of 10: \frac{3}{10}. Second draw white. One black gone; 7 white still there, out of 9: \frac{7}{9}. Third draw white. One white now gone, so 6 white out of 8: \frac{6}{8}. Multiply: \frac{3}{10}	imes\frac{7}{9}	imes\frac{6}{8}=\frac{126}{720}=\frac{7}{40}=0.175. Takeaway. Drawing the scarce color first (black, with only 3 in the box) is the hardest step, and it drives most of the small final probability.

Drawing black, white, white in order

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