# Positive probability

Assume that P(A) >0. Can one express this as follows?

1. The event A has positive probability;

2. The event A is of positive probability;

3. The probability of A is positive?

• @Chenmunka No, it can not, but it can be zero. Aug 15, 2014 at 11:31
• The title doesn't seem to reflect the content of the question. Aug 15, 2014 at 13:59
• @jinawee You were right. I changed it. Aug 15, 2014 at 14:06

In mathematics, probability is expressed on an interval from zero to one, inclusive. A zero probability means the event will never be realized; one means that it's assured. Positive means strictly greater than zero, and is a valid way of characterizing a probability. Another possibility is nonzero (optionally hyphenated). Because of the definition of probability, both positive and nonzero convey exactly the same information.

Judging by this Ngram and these three searches on MathOverflow (the SE site for professional and research level mathematics), positive is currently the preferred term, though it wasn't more than about forty years ago. Both are valid and immediately and unambiguously understood by someone familiar with the relevant mathematics.

Regarding the grammar of your sentences, all three are strictly correct. However, number two is quite archaic sounding and not at all common. Number one is, generally speaking, the most common choice, because we usually speak of events as possessing probabilities, placing the emphasis on the action. Of course, there are plenty of situations where you would want to make the probability the primary agent, and there's nothing wrong with sentence three.

Another extremely common construction is event X with probability Y, though this doesn't serve the same grammatical functions as your examples.

From reading math books, I know that 3 is possible.

Ngrams not found: is of positive probability The Ngram Viewer is case sensitive. Check your capitalization!

suggests that 2 is unusual and likely unused. I've never seen it either.