**Probability theory** is an important branch of mathematics which is quite familiar to the students who study mathematics in higher classes. This branch deals with happening of things. **For Example:** Weather forecast of some area says that there are twenty percent probability that it will rain tomorrow. The probability is said to be the chances of some event to be happen.

The term "

More elaborately, in the branch of probability, an event is defined to be the set of all the possible outcomes for an experiment.

On the basis of quality events, these are classified into three types which are as follows:

Let us study them one by one in detail in the sections below.

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As the name itself suggests that these events are independent to one another. The two or even more events, each of whose outcomes are independent of the outcomes of others, are said to be independent events. These are the events which do not affect one another. The independent events do not depend on something that already occurred previously. These events are independent because there is no affect of other events on them.

**For Example:**

In a toss of fair coin, the event "head" (or "tail") is perfectly independent event. Each toss is said to be an independent event; since getting a head or tail is independent to what came up before. Every toss of a coin is isolated from the previous attempts.

Getting a particular number on the rolls of a die is also an independent event, since every roll is not affected by previous one.

Let us suppose there are two event X and Y. These are said to be independent events if and only if

**P(X $\cap$ Y) = P(X) . P(Y)**
**Two independent events can be illustrated by the following Venn diagram:**

Dependent events are those which depend upon what happened before. These events are affected by the outcomes that had already occurred previously. i.e. Two or more events which depend on one another are known as dependent events. If the one event is by chance changed, then another is likely to be differ.

**For instance:**

**i)** Lets say three cards are to be drawn from a pack of cards. Then the probability of getting an ace is highest when first card is drawn, while probability of getting an ace would be less when second card is drawn. In the draw of third card, this probability would be dependent upon the outcomes of previous two cards. We can say that after drawing one card, there will be less cards available in the deck, therefore the probabilities tend to change.

**ii)** Suppose that we want to have a queen. With the first draw of card, the chances of getting a queen are 4 out of 52 cards. If we get a queen in first draw, then the probability of getting queen in second draw will be 3 out of 51 cards. Thus, these are said to be the dependent events, since the probability of second event depends on the outcome of the first draw.

Mutually-exclusive events are those events which cannot happen at the same particular time. We can say that the events which possess no common outcome are known as**mutually exclusive events**. Lets say P and Q be two events. These are said to be mutually exclusive if event P happens and event Q cannot; also, if event Q happens and event P cannot. The events "it will rain today" and "it will not rain on today" are two mutually-exclusive events.

In other words, two or more events are defined as mutually-exclusive events, when occurrence of any one indicates that the other one will not happen at the same time. Mutually-exclusive events are also known as**disjoint events**.

**It is also important to distinguish between independent and mutually exclusive events. Independent events are those which do not depend on one another; while mutually exclusive events cannot occur together at one time.**

**For Example:**

**i)** On a throw of a die, the two events "getting 1" and "getting 5" are two mutually-exclusive events because we will never get 1 and 5 both a one time in a throw.

**ii)** Getting a head or a tail are two mutually-exclusive events since they cannot occur in one throw.

**iii)** Drawing a king or a queen are mutually-exclusive events because both cannot be drawn in one time as shown in the following Venn diagram.

The two events "having an ace" and "having a spade" are**not** mutually exclusive since we may even draw an "ace of spade". So, these two events can occur in the same draw.

For two events X and Y, we have

P (X $\cup$ Y) = P(X) + P(Y) - P(X $\cap$ Y)

If in case X and Y are mutually exclusive events, then there will be no common event. So

P(X $\cap$ Y) = 0

Hence, we have

**P (X $\cup$ Y) = P(X) + P(Y)**i.e. we may say that for two events are mutually exclusive, the probability of one or other event is equal to the sum of their separate probabilities.

In a toss of fair coin, the event "head" (or "tail") is perfectly independent event. Each toss is said to be an independent event; since getting a head or tail is independent to what came up before. Every toss of a coin is isolated from the previous attempts.

Getting a particular number on the rolls of a die is also an independent event, since every roll is not affected by previous one.

Let us suppose there are two event X and Y. These are said to be independent events if and only if

Dependent events are those which depend upon what happened before. These events are affected by the outcomes that had already occurred previously. i.e. Two or more events which depend on one another are known as dependent events. If the one event is by chance changed, then another is likely to be differ.

Mutually-exclusive events are those events which cannot happen at the same particular time. We can say that the events which possess no common outcome are known as

In other words, two or more events are defined as mutually-exclusive events, when occurrence of any one indicates that the other one will not happen at the same time. Mutually-exclusive events are also known as

The two events "having an ace" and "having a spade" are

For two events X and Y, we have

P (X $\cup$ Y) = P(X) + P(Y) - P(X $\cap$ Y)

If in case X and Y are mutually exclusive events, then there will be no common event. So

P(X $\cap$ Y) = 0

Hence, we have

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