Salomon Rondon gave the Baggies the lead in the 16th minute with his ninth goal of the season, before goalkeeper Artur Boruc tipped a Craig Gardner penalty onto the bar at the end of the first-half.
It looked like that miss wouldn't matter as Albion closed in on their eighth 1-0 Premier League win of the season.
But having probed away at Tony Pulis' defence for the majority of the match, Bournemouth eventually equalised from a simple long throw, flicked into Ritchie's path by Claudio Yacob.
Craig Dawson hit the post in the dying minutes as the search for a win continues into its third month.
The draw keeps both teams level on points, and leaves Albion needing a victory next Sunday at home to Liverpool if they're going to better last season's tally of 44.

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