LAS VEGAS, Nev. (AP) _ Independent race and sports books, which pioneered race and sports betting in Nevada, now face extinction because of increased competition from hotel- casinos and a loss of faith by bettors nervous over the abrupt recent closure of two independent operations.

In an industry that is generally thriving, only four race and sports books not connected with hotel-casinos remain in the state - and some of those are wondering how long they can stay in business.

''I think it's just a matter of time before we have to throw the sponge in,'' said Harry Gordon, who has operated the Churchill Downs book in Las Vegas the past 18 years. ''We just can't compete with the hotels anymore.''

Along with increased competition from the hotel-casinos, the independents were dealt a hard blow with the closing of two books in the past two months that left bettors holding worthless win tickets.

Gamblers also lost access to hundreds of thousands of dollars they had in phone accounts at Gary Austin's and the Santa Anita books, scaring others who have money on deposit at the remaining independent books.

''After the news hit about Santa Anita, people that have telephone accounts here began withdrawing 50 to 70 percent of their money,'' said Vic Salerno, who operates Leroy's Race and Sports Book. ''It's created a lack of confidence with people. How many times do you have to get burned before you wise up?''

The two recent closures, one attributed to a robbery and the other to overwhelming debts, added to last year's closing of the Hollywood Race and Sports Book, almost halved the few independent books still in operation. Besides Leroy's and Churchill Downs, only the Del Mar in Las Vegas and the Turf Club in Reno remain.

Ironically, until the federal wagering tax was slashed from 10 percent in 1975, and state gaming authorities decided to allow books in casinos, the independent books flourished as the only place in the country to legally bet sports.

''As soon as the 10 percent tax went down, they went into business,'' Gordon said. ''We suffered all those years paying 10 percent, then they started into it.''

Gordon said his book, which once had a strong flow of tourists, now relies almost entirely on local players.

''We can't offer them the comps for the food or shows or drinks, which the hotels do,'' he said. ''I can't blame the people for going to the hotels. They get a lot more there.''

This season has been particularly tough for Churchill Downs, says Gordon, even before the closings that have begun to cut into his phone account business.

''It hasn't been a good season because the action is spread out so much,'' he said. ''How long we continue in operation depends only on how long we want to work for nothing.''

An expanded book at Caesars Palace and the addition of books at the MGM Grand and Hilton hotels make it easy for tourists to wager without having to leave their resorts and go looking for a place to bet.

''It gets very hard to compete with the major hotels,'' Salerno said. ''The only thing you've got is accessability and courtesy. Of course, they always used to get paid, too.''

Salerno said Leroy's has had a good year so far and is healthy financially. Still, he said, the fixed costs of staying in business have increased dramatically.

''We're offering simulcasting of horse races at a lot of added cost and the return hasn't been anywhere near what we thought it would be,'' he said. ''All the other costs are up, too.''

While major hotel-casinos may hope their books can just break even and justify their existence with extra business brought in the resort, the independent books have to make their money on bets taken against some of the smartest gamblers in the world.

''If you make one mistake in your odds, it can cost you several thousand dollars,'' Salerno said. ''Even in the best of times, it's a tough business. You got all the smart guys out there against you.''

End Adv Tues PMs Dec. 24