From the Wikipedia Page for Let's Make a Deal:
Prizes generally are either a legitimate prize, cash, or a "Zonk".
Legitimate prizes run the gamut of what is typically given away on
game shows, including trips, electronics, furniture, appliances, and
cars. Zonks are unwanted booby prizes (e.g., live animals, large
amounts of food, fake money, fake trips or something outlandish such
as a giant article of clothing, a room full of junked furniture,
etc.). Sometimes Zonks are legitimate prizes but of a low value (e.g.,
Matchbox cars, wheelbarrows, T-shirts, grocery prizes, etc.). On rare
occasions, a trader appears to get Zonked, but the Zonk is a cover-up
for a legitimate prize.
Though usually considered joke prizes, traders
legally win the Zonks. However, after the taping of the show, any
trader who had been Zonked is offered a consolation prize (currently
$100) instead of having to take home the actual Zonk. This is partly
because some of the Zonks are intrinsically or physically impossible
to receive or deliver to the traders (such as live animals or the guy
in an animal costume), or the props/employees are owned by the studio.
A disclaimer at the end of the credits of later 1970s episodes read
"Some traders accept reasonable duplicates of Zonk prizes."
To back this up, this is from an interview with Monty Hall, one of the first presenters:
Interviewer: People who got stuck with the "zonks," the joke prizes, were they
good-natured about losing out?
Monty Hall: In 4,700 shows, I got kissed 50,000 times. Even when they
lost, they were very nice about it. But you know the law in game shows
- if you go on a show and you win a donkey, that's your prize. You're entitled to it. So if a person won one of our zonks, they could take
it home. But in 99 percent of the cases, we would offer them something
after the show - a washer and dryer or a color TV or something,
instead of that very valuable zonk, and they would take it. In 1
percent of the cases, they didn't.
There was a time when a farmer won five calves and he wanted the
calves. That cost me a fortune because when you rent them from the
animal place, they're expensive. And there were other cases like that.
Like people won dogs; they would keep dogs. They wouldn't keep cats.
They would keep dogs. It was a very genial atmosphere, over 27 years.
So despite the "zonks" being joke prizes, people did indeed take them home on rare occasions. I'm sorry to say I haven't found an example of a goat being taken home, but I'm hoping a calf will do!