I heard this story today on Science Friday (a cool program that runs on NPR). If you go to the link you can listen to the story, but it's basically an interview with the Technical Lead, Space Shuttle Probabilistic Risk Assessment and the Space Shuttle Program Risk Manager who both, as you might suspect, have to do a bit of math in their jobs.
The gist of the story is that, through new techniques and with new knowledge that we have now, the risk of catastrophic failure in the early space shuttle missions was much, much higher than we thought. NASA sometimes said that the risk was something like 1 in 100,000, and even after the first shuttle disaster (Challenger), the report said the risk was something like 1 in 100. This latest study has now concluded that the risk in those first 25 flights (Challenger was the 25th flight) was more like 1 in 10, which means there was only a 6% chance of making it to flight 25 (Challenger) without a catastrophic accident. (The math: They've calculated a 1 in 9 chance of failure for the first 9 flights which means about 88.8% chance of success; 1 in 10 for the next 16 flights which means a 90% chance of success. That works out to about 89.5% chance of success over those 25 flights. 89.5% raised to the 25th power equals roughly 6% chance of all of them being successful).
The image above is from a really interesting document (pdf) if you're interested, which lists the estimated risks over time (first page), and the specific risks they've calculated (pages 2 - 5). Lots and lots and lots of probability here.
The good news (sort of) is that they estimate the risk now is only 1 in 90 (went down significantly after Challenger, and then again after Columbia). I say "sort of" because that still seems pretty darn high to me. What do you think? Do you think a 1 in 90 chance of losing the shuttle and crew is worth it to explore space?
Update: Cool video of latest launch. This is multiple videos in one - the first 7:50 is from a camera on the solid rocket booster (from liftoff to splash down), then a new video starts from a different perspective.