How does the definition of probability account for differing chances of an event occurring? My teacher defined that the probability of an outcome is defined as the:
${\,Number\,of\,specific\, outcomes \over\,Number\,of\,total\, outcomes}$
This definition makes sense to me with a normal coin. The probability of 3 heads out of 5 tosses is simply:
$\binom{5}{3}\over 2^5  $
The numerator is all the combinations where Heads comes up 3 times. And the denominator is all the possible outcomes of flipping a coin 5 times. Makes sense!
BUT this formula falls apart as soon as I try this with a weighted coin. Let's say Heads comes up 80% of the time. The numerator should still be the "Number of specific outcomes", aka 5 choose 3. This means the "Number of total outcomes" is no longer 32, even though only one of two possibilities occurs each trial.
Is there a way to conceptually upgrade this definition of probability to account for my weighted coin example?
 A: Indeed, that formula is only valid when all outcomes have equal measure (or 'weight'). Otherwise you do have to have some method to account for the bias : ie give each outcome a weight so the probability for an event equals a ratio of weighted sums.
$$\text{Probability}=\dfrac{\text{Weighted count of outcomes in event}}{\text{Weighted count of outcomes in sample space}}$$
A: You can write down a list of axioms that probabilities should satisfy. Usually before stating the axioms of probability, we first express basic probability concepts in the language of sets.
If $S$ is the set of all possible outcomes our random experiment, then an "event" is a subset of $S$. For example, if the random experiment is flipping a coin twice, then $S = \{ HH, HT, TH, TT\}$. The event "got the same result in both coin tosses" is the subset $\{ HH, TT \}$ and the event "flipped at least one heads" is the subset $\{HH, HT, TH \}$. To say that an event $E \subset S$ "occurs" means that when we perform our random experiment, the outcome of the experiment is an element of $E$.
Now that we have agreed that events are subsets of $S$, we can note that $A \cap B$ is the event that $A$ occurs and also $B$ occurs. Similarly, $A \cup B$ is the event that $A$ occurs or $B$ occurs.
We are ready to state our axioms. If $P(E)$ is the probability of event $E$ occurring, then the following conditions should be satisfied:


*

*$0 \leq P(E) \leq 1$ for any event $E$.

*$P(S) = 1$.

*If $A$ and $B$ are disjoint events, then $P(A \cup B) = P(A) + P(B)$. (In fact, if $\{A_j\}$ is a sequence of pairwise disjoint events, then $P(\cup_j A_j) = \sum_j P(A_j)$.)


We can think of $P(E)$ as being our prediction of the fraction of trials in which the event $E$ will occur if we repeat our random experiment a large number of times. The axioms of probability given above can be viewed as being rules that must be satisfied in order for our predictions to be consistent with one another. For example, when rolling a loaded die, if we predict that $1/10$ of the time we will get a $1,$ and $3/10$ of the time we will roll a $2$, then for the sake of consistency we must predict that $4/10$ of the time we will roll either a $1$ or a $2$.
In situations where not all outcomes are equally likely, the above axioms can be used to calculate probabilities of various events. This is the approach taken in most probability textbooks, such as A First Course in Probability by Sheldon Ross (which you might like reading).
