Let's reason through this problem:

1. We know there are 100 people in total, with 50 wearing red hats and 50 wearing blue hats.
2. The probability of selecting a person with a red hat on the first pick is 50/100 (since there are 50 red-hat wearers out of 100 people).
3. Now, after picking one person with a red hat, there are now 49 red-hat wearers left among the remaining 99 people.
4. So, the probability of selecting another person with a red hat on the second pick is 49/99.
5. To find the overall probability of both picks being red-hat wearers, we multiply the probabilities of each individual event occurring: (50/100) * (49/99).

Hence, the probability of both selected people wearing red hats is (50/100) * (49/99), which simplifies to 2450/9900 or 7/29.

Final answer: The probability is 7/29.
There are 100 people in a room. 50 of them are wearing red hats, and the other 50 are wearing blue hats. If we randomly select two people from the room, what is the probability that both of them will be wearing red hats?