"prone to believing" is correct and usable in written English.
It means being inclined or likely to believe something. Example: She was always prone to believing rumors and gossip without questioning their validity.
But we are also prone to believing that the last great immigrants were the ones who arrived decades ago.
It is also why we are prone to believing crazy things about what the other "side" believes.
The Afghan president, Hamid Karzai, is erratic, emotional and prone to believing paranoid conspiracy theories, according to frustrated diplomats and foreign statesmen.
He may be vital to western plans in Afghanistan but Hamid Karzai is regularly described by frustrated diplomats and foreign statesmen as erratic, emotional and prone to believing paranoid conspiracy theories.
The police chief trained drummers to use them as a way to gain trust and reach villagers most prone to believing fake news.
By removing the limitations imposed by geography, Jackson said, social media has made "it easier for people who might be prone to believing in conspiracy theories to find one another and to converse and to develop those conspiracy theories further".
He also concluded that people are more prone to believe the "Wondrously False" than the "Wondrously True".
Older people with age-related impairments to the frontal cortex do poorly on gambling tasks and, experiments show, are prone to believe misleading advertising.
Despite what many are prone to believe -- partly because of statements by national leaders -- schools are among the safest places for children to be.
"As the conservatives have become more prominent, other students are more prone to believe that they are being indoctrinated," Schneider says.
This shows that children are more prone to believe and mimic things they are told or shown in media aimed at them.
