Okay, take this for what you will, but after looking at Kennadog's numbers, and the numbers over at the wiki (which is proabbly where kenna got them from? Yes?) this is how I think our odds turn out:
Tied for 8th/9th/10th
which if you look at the example year at the bottom of the wiki, means that the percentages for the 8th spot, 9th spot, and 10th spot, will be be added up and divided by three. Each team getting an equal %.
So:
total chance of #1:
5.6% divided by 3 = 1.87%
total chance of #2:
6.6% divided by 3 = 2.2%
total chance of #3:
7.9% divided by 3 = 2.63%
Total chance of landing a Top 3 pick =: 1.87 + 2.2 + 2.63 = 6.7%
Note, that there is a easily spottable error in the following breakdowns (i.e. they do not add up to 100%, so the fact that we are tied must tweak the percentages higher for 8th and lower for 10th (reason being that due to the tie the 10th pick this year has a higher than normal chance of going Top 3, and thus knocking the 8th place team back into 9th/10th, whereas the 8th place team due to the tie has a lower than normal chance at going Top 3. Anyway:
If we win the coin flip or whatever to be called "8th", then we have a 6.7% chance to pick Top 3, a 72.5% chance to pick 8th, 18.4% at 9th and about 1% at 10th.
If the coin flip makes us "9th", then we have a 6.7% chance to pick Top 3, a 79.7% chance to pick 9th, 12.1% at 10th and about 1% at 11th.
If the coin flip makes us "10th", then we have a 6.7% chance to pick Top 3, an 87.0% chance to pick 10th, 8.9% at 11th and about 1% at 12th.