Basic Probability Calculator

The probability of an event occurring given that another event has already occurred. It is denoted as $P(A|B)$ (read as "P A given B"), where $A$ is the event and $B$ is the condition. For example, the probability of rolling a 6 () given that the die roll is even () is:$$P(6|\text{Even}) = \frac{1}{3} = 0.333$$

The formal definition of conditional probability is given by the formula:

$$P(A|B) = \frac{P(A \cap B)}{P(B)}$$By rearranging the formula, we can obtain the joint probability of $A$ and $B$ , which is also known as the multiplication rule:$$P(A \cap B) = P(A|B) \times P(B)$$

If events $A$ and $B$ are independent, then the conditional probability of $A$ given $B$ is equal to the probability of $A$: $P(A|B) = P(A)$. Moreover, if $A$ and $B$ are mutually exclusive, then the conditional probability of $A$ given $B$ is zero: $P(A|B) = 0$.

The probability that either of two events occurs ($A$ or $B$) is the sum of their individual probabilities, minus the probability of both occurring together (if they can overlap).$$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$For example, let $A$ be the event of rolling an odd number(,,) and $B$ be the event of rolling a number greater than 4 (). The probability of rolling an odd number or a number greater than 4 is:$$P(A \cup B) = P(A) + P(B) - P(A \cap B) = \frac{3}{6} + \frac{2}{6} - \frac{1}{6} = \frac{2}{3} = 0.667$$

A combination is a selection of objects where order doesn't matter. The number of combinations of $n$ distinct objects taken $r$ at a time is denoted as $C(n,r)$ or $^nC_r$ or$\binom{n} {r}$ and is calculated as:$$C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}$$

A permutation is an arrangement of objects where order matters. For example, (1, 2) is different from (2, 1). The number of permutations of $n$ distinct objects taken $r$ at a time is denoted as $P(n,r)$ or $^nP_r$ and is calculated as:$$P(n,r) = \frac{n!}{(n-r)!}$$