For those not familiar to the right to privacy, it would technically be covered by the 9th amendement (the one no one ever seems to know
).
As for random bag checks, it is sort of a interesting toss up. If you allow the searches, it can indeed violate the rights of people and make them feel considerably less comfortable with taking that form of transportation. If you disallow them, you do have a chance of a bomb being used on the subway. Things to consider:
1. Searching every single bag would have highly impractical if not impossible.
2. One seeking to use a bomb is likely to circumvent security at the mere suspicion of investigation, making these checks relatively ineffective.
3. As mentioned before, a bomb could easily be rigged to go off when a bag is opened for inspection, enough of these, and I don't think you will see people lining up to be security for subway station.
4. Explosives generally involve combinations of chemicals that can be detected by dogs, and electronic equipment. Using these would probably be a more effective way of scanning for explosives, without violating privacy, and possibly also being able to avoid the suspects from knowing that the police are onto them.
All in all, I don't think bag checks would be effective enough to deter terrorist attacks to warrant their usage. I certainly wouldn't feel any safer.