What is the Birthday Problem?

Last Updated : 30 Dec, 2021

Probability is also known as a possibility. This means math of chance, that trade in the happening of a likely event. The value is deputed from zero to one. In math, Probability or math of chance has been shown to guess how likely affairs are to occur. Basically, the probability is the scope to which something is to be anticipated to happen.

Probability

To recognize probability more deeply, take an example as rolling a dice, the possible outcomes are 1, 2, 3, 4, 5, and 6. The possibility of occurring any of the likely affairs is 1/6. As the probability of occurring any of the events is the same so there are similar chances of getting any likely number, in this case, it is either 1/6 or 50/3.

Formula of Probability

Probability of an event = {Number of favourable affairs} ⁄ {total number of affairs}

Types of Events

There are different types of outcome based on a different basis. One type of event is an equally likely event and a complimentary event. Another type of event is an impossible and sure event. Then there are simple and compound events. There are independent and dependent events, mutually exclusive, exhaustive events, etc. Let’s understand these events in detail.

Equally Likely Events: After throwing a dice, the probability of getting any of the equally likely outcome is 1/6. As the occurring of the outcome is an equally likely outcome so there is the equal or similar possibility of prevailing any number in this case it is either 1/6 in fair dice rolling.
Complementary Events: There is a chance or possibility of only two result which is an event will occur or not. Like a person will read a book or not read a book, cleaning a kitchen or not cleaning a kitchen, etc. are examples of complementary events.
Impossible and Sure Events: If the possibility of happening an equally likely affair is 0, these types of events are known as impossible events and if the probability of occurring of a likely affair is 1, these types of events are known as a sure event. In other words, the sample space S is a sure event and the empty set ϕ is an impossible event.
Simple Events: Any event carrying one point of the sample space is known as a simple event in probability. For example, if S = {34, 28, 89, 47, 88} and E = {69} it means E is a simple event.
Compound Events: Contrary to the simple event, if any event carries more than one single space of the sample space then such an event is called a compound event. Checking the same example again, if S = {34, 28, 89, 47, 88}, E₁ = {34, 47}, E₂= {28, 34, 88} then, E₁and E₂ shows two compound events.
Independent Events and Dependent Events: If occurring of any events is completely uninfluenced by the happening of any other outcome, such events are known as independent events in probability and the events which are pretentious by other events are known as dependent events.
Mutually Exclusive Events: If the occurrence of one event keep out the occurrence of another event, such events are mutually exclusive events i.e. two events don’t have any common number. For example, if S = {11, 12, 13, 14, 15, 16, 17} and E₁, E₂ are two events such that E₁ consists of numbers less than 14 and E₂ consists of numbers greater than 17. So, E₁ = {11, 12, 13} and E₂ = {14, 15, 16, 17}. Then, E₁ and E₂ are mutually exclusive.
Exhaustive Events: A group of events is called exhaustive events, which says that one of them must happen.
Events Associated with “OR”: If two events A₁ and A₂ are associated with OR then it means that either A₁ or A₂ or both. The merging symbol (∪) is used to represent OR in probability. Thus, the event A₁ U A₂ indicates A₁ OR A₂. If one has mutually exhaustive events A₁, A₂, A₃… an associated with sample space S then, A1 U A2 U A3 U … An = S
Events Associated with “AND”: If two events E₁and E₂are related with AND then it states that the connecting of components is similar to both the events. The intersection symbol (∩) is used to represent AND in probability. Thus, the event E₁ ∩ E₂ shows E₁ and E₂.