Basic Probability Calculator

Created:September 10, 2024

This calculator helps you compute the probabilities of conditional events, unions, combinations, and permutations. It also explains the concepts of basic probability with examples using dice rolls.

Calculator

Calculates P(A|B) and/or P(B|A) given P(A) and/or P(B), and P(A∩B). Enter as many values as you have.

P(A)

P(B)

P(A∩B)

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Conditional Probability

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:

Even numbers highlighted, 6 emphasized

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: $P(A|B) = 0$ .

Addition Rule (Union)

The probability that either of two events occurs (

A

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 ().

First Die: rolling an odd number

Second Die: 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

Combination

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)

^nC_r

\binom{n} {r}

and is calculated as:

C(n,r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}

For example, let's consider selecting 2 different even numbers from the set 6. The number of ways to do this is:

C(3,2) = \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3 \times 2}{2 \times 1} = 3

This means there are 3 different ways to select 2 even numbers from 2, 4, and 6:

which are:

The order doesn't matter because () is considered the same as ().

Calculating Combinations in R

In R, you can calculate combinations using the choose() function, which is equivalent to the mathematical notation $\binom{n}{r}$ or $C(n,r)$ .

choose(3, 2) # Output: 3

Permutation

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)

^nP_r

and is calculated as:

P(n,r) = \frac{n!}{(n-r)!}

For example, let's consider rolling three dice and getting different even numbers (2, 4, 6) on each die. The number of ways to do this is:

P(3,3) = \frac{3!}{(3-3)!} = 3! = 3 \times 2 \times 1 = 6

This means there are 6 different ways to arrange the even numbers 2, 4, and 6 on three dice:

which are:

The order matters because (2, 4, 6), (4, 6, 2), and (6, 2, 4) are all considered different arrangements.

Calculating Permutations in R

In R, you can calculate permutations by using a combination of factorial() functions, which is equivalent to the mathematical notation $P(n,r) = \frac{n!}{(n-r)!}$ .

factorial(3) / factorial(3-2) # Output: 6

Basic Probability Calculator

Calculator

Learn More

Conditional Probability

Addition Rule (Union)

Combination

Calculating Combinations in R

Permutation

Calculating Permutations in R

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