8/17
If you pick an ordinary coin and flip it 3 times, coming up heads each time, that's a probability of 1/10 * 1/2 * 1/2 * 1/2 = 1/80. Since you have 9 such coins the probability of doing so with a ordinary coin is 9/80. Finally, you have a 1/10 chance of picking up the double headed coin, so the probability of coming up with all heads on that coin is 1/10.
So you now have 2 probabilities for seeing 3 heads in a row.
9/80 and 1/10. Since you do see 3 heads in a row, those two probabilities have to add up to 100%. So let's see what percentage of the sum each probability comes to.
Total = 9/80 + 1/10 = 9/80 + 8/80 = 17/80
Ordinary coin = 9/80 / 17/80 = 9/80 * 80/17 = 9/17
Double headed coin = 8/80 / 17/80 = 8/80 * 80/17 = 8/17
The probability of having picked up the double headed coin and observing 3 heads in a row is 8/17
