Are "Probability Distribution" and "Probability Function" the same thing? I have these two definitions for "Probability distribution" and "Probability function":

Probability Distribution: Assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. (Wikipedia)

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Probability Function: Assigns an event A a probability P(A) that represents the likelihood of event A occurring. (MITx 6.00.2x Introduction to Computational Thinking and Data Science)

Those are very similar yet slightly different definitions. Specifically, Probability Function seems to refer to a single event whereas Probability Distribution seems to refer to a collection of events. I've googled around but not found a satisfying answer as it seems that both terms are used in different contexts.
 A: My reading is that they are not the same thing. The following paragraph from the Wikipedia page for "probability density function" may help or may just confuse you further:
The terms "probability distribution function"[1] and "probability function"[2] have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the "probability mass function."
Your definition of a "probability function" ("assigns a probability to an event") seems more applicable to a probability mass function, $f(x) = Pr(X=x)$, which is a function that gives the probability that a discrete random variable is exactly equal to some value. A probability density function, maps a continuous variable to the relative likelihood of that value as compared to other values. For given $x$, the value of the probability density function $f(x)$ may have a value greater than 1, which is a no-no when it comes to probability. For example, the uniform distribution on the interval [0, .5] has a value of 2 on that interval and 0 elsewhere. In order to obtain probability from the probability density function, one must integrate between the values of interest. This means, interestingly, that the probability of a given specific $x$ occurring is technically zero. This is because if in integrate $f(x)$ from $a$ to $b$ where $a=b$ you get $F(a)-F(a) = 0$ where $F(x)$ is the cumulative distribution function.
