Snapped

For all three of your family members to not be included in the snap--sorry, mad science experiment--you all need to be in the $50$ people who survive. There is a $\frac{50}{100}$ chance you will survive, but there is only a $\frac{49}{99}$ chance one of your family gets in (because you've already taken the $100$ th spot, now there are technically $99$ people in the community. It's a bad way of thinking about it, but it works. Each person does not have a $\frac{1}{2}$ chance). There is a $\frac{48}{98}$ chance your last family member gets in. Multiplying all of this together, you get $\frac{(50)(49)(48)}{(100)(99)(98)}=\frac{4}{33}$ chance of your family of three surviving.

People may make the mistake of thinking everyone has a 0.5 chance of surviving. But if that is the case, there is a chance, albeit extremely small, that all 100 people survive or all get wiped out. With that line of thought, the chance for all 3 of you surviving is 0.125 or $\frac{1}{8}$ .

To make sure you all survive, this guy must choose 50 people from 97 other people. He has $\frac { 97! }{ 47!\times 50! }$ choices

If he doesn't have to make sure you all survive, he can choose from 100 people. So he has $\frac { 100! }{ 50!\times 50! }$ choices

So the chance you all survive is $\frac { 97! }{ 47!\times 50! } \frac { 50!\times 50! }{ 100! } =\frac { 4 }{ 33 }$