statistics/probability help!

[remo]

fair 6-sided dice is thrown together with a biased dice. The biased dice has the probability of a 6 being thrown is 1/3. Given that only one 6 was rolled, what is the probability that is was rolled on the biased dice? the answer is 5/7. Why? Help!

[StuartS]

You know that a 6 was thrown So this could be on the biased die or the fair die.

P(of one 6) = 1/3 x 5/6 + 2/3 x 1/6
= 7/18

However the probability of getting one 6 on the biased die only and not on the fair die is
1/3 x 5/6 = 5/18

Now compare the two probabilities to give

P(the die is biased given one 6 was thrown) = 5/18 / 7/18 = 5/7