Ultimately, any insurance company who has to pay anything in this scenario will go back after the guy in Car #3, who caused the accident, and sue him for his 401k, pension, house, and other financial assets, including his future earnings.
In the meantime, however, you are going to be brought into the lawsuit, and the only portion of your car insurance that will help you pay for the injuries the lady in Car #2 sustained, is the Uninsured Motorists portion of your car insurance.
Your Bodily Injury insurance, on your car insurance, will not pay, because you were not responsible for the accident.
If you have coverage levels of at least $100,000/$300,000 for your Uninsured Motorists coverage, your insurance company will cover the $75,000 in injuries that the lady in Car #2 sustained.
Derrick Rupp and Michelle Brill both had answers that focused only on Uninsured Motorists coverage, and Michelle's name was drawn as the $50 winner.