# Probability Calculator

This probability calculator computes the probability of two independent events for various situations. It also provides the solving process. To use the calculator, please select the data you know about the two independent events first, input their values, then click the "Calculate" button.

## Probability of events

The probability of an event is the likelihood of an event occurring. One classic example is the flip of a coin. When a fair coin is flipped, it has an equal likelihood, or probability, of landing on either heads or tails. An event that is sure to occur has a 100% probability, while an event that can never occur has a 0% probability. Thus, the probability of an event ranges between 0-100%. Probabilities are often expressed as decimals between 0 and 1, which can easily be converted to percentages by multiplying by 100.

The probability of a given event occurring can be computed using the general formula:

Referencing the example of a flip of a coin, there are two possible outcomes (heads or tails). If we want to determine the probability of one flip of a coin resulting in an outcome of heads, there is only 1 way for this event to occur: the coin must land on heads. Thus, the probability of the coin landing on heads after 1 flip of the coin is computed as follows:

The probability of the coin landing on tails is also the same. While this example is relatively simple, probability calculations get more complicated with different types of events, and it is not always as easy to determine the values for the components of the above formula.

Two of the key types of events discussed in probability are independent and dependent events.

### Independent events

An independent event is an event in which the outcome of the event is not affected by previous events. The flip of a coin is an example of an independent event. Each time a coin is flipped, there is a 50% chance that it lands on either heads or tails. If the coin lands on tails, and the coin is then flipped again, the probability of the coin landing on heads or tails remains the same. In other words, each individual flip of the coin has no effect on any other flip of the coin. Even if the coin were flipped 1000 times, and it somehow landed on tails 999 times, there would still be a 50% chance of the coin landing on either heads or tails on the 1000^{th} flip.

The probability of an independent event is calculated as shown above using the formula:

The probability of multiple independent events is calculated by multiplying the probabilities. For example, the probability of 5 tails occurring in a row is the product of the probability of tails occurring on each flip of the coin. Since we know that the flip of a coin is an independent event, the probability of each the coin landing on tails on each individual flip is 0.5, and the probability of the coin landing on tails 5 times in a row is:

0.5×0.5×0.5×0.5×0.5 = 0.5^{5} = 0.3125 = 3.125%

### Dependent events

A dependent event is one in which the probability of a subsequent outcome is affected by the result of the previous outcome. For example, consider a bag that contains 5 coins, 3 of which are quarters and 2 of which are nickels. If one coin is removed from the bag randomly, there is a chance of the coin being a quarter, and a chance of the coin being a nickel. If the coin is not replaced, and another coin is removed from the bag, it is not possible to determine the probability of drawing a nickel or quarter without knowing what coin was first drawn. If the first coin drawn were a nickel, there is a chance of drawing a nickel and a chance of drawing a quarter on the second draw. If the first coin drawn from the bag was a quarter, there would be a chance of drawing either a nickel or a quarter on the second draw. Thus, the outcome of the 2^{nd} draw is dependent on the outcome of the 1^{st}.

Note that in the example above, if the removed coin were replaced after each draw, then the event described would be an independent event, since the probability of drawing either a quarter or nickel would be the same for each trial.

## Probability notation and calculation of two independent events

In probability, events are expressed using capital letters, such as A and B. The probability of some event, A, is expressed as P(A), and the probability of another event, B, is expressed as P(B). There are a number of other symbols used to indicate different types of probabilities or events. Below is a list of the different notations used by this calculator, as well as the formulas used to calculate each.

**P(A)**—probability of event A. This is the probability of a given event, A, occurring.

**P(B)**—probability of event B. This is the probability of a given event, B, occurring.

**P(A∩B)**—probability of the intersection of A and B. The ∩ symbol is read as "intersection" or "AND." The probability of the intersection of A and B can therefore also be read as the probability of A AND B. This is the probability of elements being in both A and B. For example, given two sets of numbers A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, the intersection of sets A and B is A∩B = {3, 4, 5} because those are the three elements present in both A and B. The formula for calculating the probability of the intersection of A and B is:

P(A∩B) = P(A) × P(B)

**P(A∪B)**—probability of the union of A and B. The ∪ symbol is read as "union" or "OR." The probability of the union of A and B can therefore also be read as the probability of A OR B. This is the probability of elements being in either A or B. In other words, it is represented by all the elements of both sets. For example, given two sets of numbers A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, the union of sets A and B is A∪B = {1, 2, 3, 4, 5, 6, 7}. The formula for calculating the probability of the union of A and B is:

P(A∪B) = P(A) + P(B) - P(A)P(B)

**P(AΔB)**—probability of the symmetric difference of A and B. This is also referred to as the disjunctive union of A and B. The disjunctive union of A and B is the set of elements that occur in either set A or B, but not in their intersection. For example, given two sets of numbers A = {1, 2, 3, 4, 5} and B = {3, 4, 5, 6, 7}, the symmetric difference of A and B is AΔB = {1, 2, 6, 7}. The formula for calculating the probability of the disjunctive union of A and B is:

P(AΔB) = P(A) + P(B) - 2P(A)P(B)

**P(A')**—probability of the complement of A. The ' symbol indicates the complement of the set. This is the probability of all the elements that are not in A. If A is a set of numbers within the set of all whole numbers between 0 and 10 such that A = {0, 1, 2, 3}, then its complement is A' = {4, 5, 6, 7, 8, 9, 10}. The probability of the complement of A is computed using the formula:

P(A') = 1 - P(A)

**P(B')**—probability of the complement of B. The ' symbol indicates the complement of the set. This is the probability of all elements that are not in B. If B is a set of numbers within the set of whole numbers between 30 and 40 such that B = {36, 37, 38, 39, 40}, then its complement is B' = {30, 31, 32, 33, 34, 35}. The probability of the complement of B is computed using the formula:

P(B') = 1 - P(A)