First one's red, will all three be?

Question

A bag holds 4 red tokens and 6 blue tokens. You draw 3 tokens one at a time without replacement. You glance at the first token and see that it is red.

Given that the first token is red, what is the probability all three tokens are red?

Accepted Answer

Condition on the first token being red and re-set the bag: after removing one red, there are 9 tokens left, 3 red and 6 blue. You now need the next two draws to both be red. P(	ext{2nd red}\mid 	ext{1st red}) = \frac{3}{9},\qquad P(	ext{3rd red}\mid 	ext{first two red}) = \frac{2}{8}. Multiply: P(	ext{all three red}\mid 	ext{1st red}) = \frac{3}{9}	imes\frac{2}{8} = \frac{6}{72} = \frac{1}{12} \approx 0.083. Takeaway. Conditioning on early draws simply updates the bag's contents; carry that reduced composition forward and chain the remaining conditional probabilities. Without replaceme…

First one's red, will all three be?

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