Soundproof Booth

Some trivia game shows ask questions to one contestant at a time instead of pitting them against each other directly with a buzzer. For example, Alice will answer a set of questions, and then Bob will answer a set afterwards.

The problem with this is, if Alice and Bob are asked different questions, it doesn't feel fair, and if they are asked the same questions, then Bob will know the questions ahead of time when they're asked to Alice.

This is where the sound-proof booth comes in. Bob is placed in a large glass booth through which no sound can enter, while Alice answers the questions. Bob usually wears headphones as well, just in case. The host is careful to let us know the purpose of the booth every time it's used, even though it's usually pretty self-explanatory. An odd tendency is for the host to whisper this fact to us, seemingly simulating what it sounds like to the person in the sound proof booth (in which case it isn't very sound proof at all).

As to why they couldn't just put Bob backstage: a sound-proof booth is just cooler.

The sound-proof booth has also been used in beauty pageants, including those in the Miss Universe organization, when the contestants are asked a final question before the judges decide the winner.

Examples of Soundproof Booth include:

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The Alex Trebek-hosted quizzer |Double Dare (not to be confused with the |kids' show).
Win Ben Steins Money: In this case, the booth is open; the headphones do all the work. Ben always got a fancy library-themed booth, while the challenger got a plain looking beige one.
On Whew!, while the blocker placed blocks on the board, the charger sat behind a scenic flat, wearing headphones that played white sound.
Cheerfully parodied in Good News Week: Despite the insisting of Paul McDermott and the elaborate fanfare, the sound-proof booths used in the show are anything but.
The Simpsons did a parody in a Halloween episode where the space alien Kang is revealed to be Maggie's biological father (long story), and the family ends up telling their story on the Jerry Springer show. When Jerry informs the family that they've had Kang sitting in a sound-proof booth backstage the entire time, Kang loudly proclaims, "I HEAR ALL!"
Australian sketch comedy show The Micallef Program parodied this, with Micallef (as the host) and Wayne (as one of the contestants) tricking the other person into the booth under the guise of a Game Show sketch, then using the opportunity to talk about him behind his back. After Wayne accused the man of being a thief, he got to chose a gas to be released into the chamber. He chose chloroform.
Played with on one episode of Home Improvement: Tim makes a booth that is anything but soundproof, but claims it to be soundproof, complete with a convincing "demonstration" using some good acting. He them prompts Al to get in the booth and make a fool of himself thinking nobody could hear him.
A comedian at Just for Laughs used the sound-proof booth in one of his routines. He mocked the whispering of "The contestant is in a sound-proof booth," saying that if it is sound-proof, the host should be yelling to prove it.

"Sound-proof booth! This is the fattest lady we've ever had on the show!"

Family Feud, during its original run, used a soundproof booth for the second contestant of "Fast Money." In the current series, sound-blocking headphones are used for this purpose.
Friend or Foe put its teams in soundproof booths so that they could only hear Kennedy and not the other teams.
The UK gameshow Mr And Mrs tested couples on how well they knew each other. One of them would go in a Sound Proof Booth, and the other would be asked questions about their spouse's likes and dislikes. Then the spouse would be taken out of the booth and asked the questions about themselves. If they matched, you got points.
The Bonus Round on The One Million Chance of a Lifetime was played in one that was wired so the couple could hear only the host.