The World Health Organization probiotics definition is "live microorganisms which when administered in adequate amounts confer a health benefit on the host."
Or check out what are probiotics for a less formal probiotics definition.
The "live microorganisms" that the WHO refer to are the good bacteria in your yogurt or the Brand X supplement you bought, and "the host" is you. And the two of you are going to get along just fine because the human body is not only designed to perform whilst carrying a cargo of these little critters, but probiotics are essential for your good health.
When scientists deliberate the definition of probiotics they like to add a few more riders for a bacteria to qualify.
There are other requirements that could be asked of our probiotic - things like its ability to adhere to intestinal cells and resistance to bile and gastric juices - but these other requirements need only be met by some probiotics. If a probiotic's task is to confer a health benefit in the mouth or throat, it really doesn't matter whether they enjoy the hurricane winds that might whistle through your bowel!