Probability

**Category : **10th Class

**Probability**

**Random experiment:**An experiment in which all possible outcomes are known and the exact outcome cannot be predicted in advance is called a random experiment.

**e.g.,** (1) Tossing a coin.

(2) Rolling an unbiased die.

**Sample space:**

The set S of all possible outcomes of a random experiment is called the sample space.

**e.g.,** (1) In tossing a coin, sample space (S) = {H,T}.

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- (2) In rolling a die, sample space (S) = {1, 2, 3, 4, 5, 6}.

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**Probability:**Probability is a concept which numerically measures the degree of certainty of the occurrence of events.

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**Definition of probability:**in a random experiment, let S be the sample space arid let E be an event. Then probability of occurrence of \[E=P(E)=\frac{n(E)}{n(S)}\],where

n(E) is the number of elements favorable in E and

n(S) is the number of distinct elements in S.

**Note : **1.** \[0\le P(E)\le 1\]**

- If \[P(E)=1\], The event E is called a certain event and if \[P(E)=0\], the event E is called an impossible event.

**Types of events:**

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**(i) Simple event or elementary event:**An event is called a 'simple event; if it is a single- ton subset of the sample space's'.

**e.g..** When a coin is tossed, sample space S = {H,T}

Let A = {H} = The event of occurrence of head and

B = {T}= The event of occurrence of tail.

Then 'A' and 'B' are simple events.

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**(ii) Mixed event or compound event:** A subset of the sample space 'S' which contains more than one element is called a compound event.

**e.g..** When a die is thrown, sample space S = {1, 2, 3, 4, 5, 6}.

Let A = {1,3,5} = The event of occurrence of odd number and

B = {5,6} = The event of occurrence of a number greater than 4.

Then, A' and 'B' are compound events.

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**(iii) Sure event:** If a random experiment 'E' has a discrete sample space 'S’; then 'S' itself is an event (E = S) called the sure or certain event of 'E'.

**e.g..** Getting a head or a tail in a single toss of a coin is a sure event.

The probability of a sure event (or certain event) is 1.

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**(iv) Impossible event:** The empty subset \['\phi '\]of 'S' \[(E=\{\})\]is called the impossible event or null event of 'E'.

**e.g..** Getting a head and a tail. both in a single toss of a coin is an impossible event.

The probability of an impossible event is 0.

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**(v) Complementary event:** For a random event 'A' of a random experiment 'E; the event complementary to 'A' is the event that "A does not occur" It is denoted by A' or A'= or A.

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**(vi) Equally likely events:** Events are said to be equally likely when there is no reason to expect any one of them rather than any one of the others.

**e.g..** When an unbiased die is thrown, all the six faces 1, 2, 3, 4, 5 and 6 are-equally likely to come up.

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**Results based on the definition of probability:**The following results are direct consequences of the definition of probability.

**(i)** If 'E' is an event of sample space's', then\[0\le P(E)\le 1\].

P(E) = 0 if and only if 'E' is an impossible event and

P(E) = 1 if and only if 'E' is a certain event.** (ii)** If 'E' is an event of sample space 'S' and 'E' (or I) is the event that E does not happen, then \[P(E')=1-P(E)\].

- Odds in favour and odds against: Let 'A' be an event of an experiment 'E'. Then, the ratio P(a) :P(A') is called the odds in favour of A and the ratio P(A'): P(a) is called the ‘odds against' A.

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**Combination of two events:**

**(i) Union of events: **

If A and Bare two events of the sample space S, then \[A\cup B(orA+B)\]is the event that either A or B (or both) take place.

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**(ii) Intersection of events:**

If A and B are two events of the sample space 'S’; then \[A\cap B\] (or AB) is the event that both A and B take place.

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**(iii) Mutually exclusive events:**

Two events A and B of the sample space S are said to be mutually exclusive if they cannot occur simultaneously. In such a case \[A\cap B\]is a null set.

**e.g..** When two coins are tossed the number of elementary events is 4 and they are (H, H), (H, T), (T, H), (T, T).These are mutually exclusive.

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**(iv) Exhaustive event:**

Two events A and B of the smaple space S are said to be exhaustive if \[A\cup B=S\], i.e.,** \[A\cup B\]**contains all sample points.

**e.g..** In tossing a coin, there are two exhaustive elementary events.

They are head and tail.

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**Note: **

**(a)**A and A' are mutually exclusive as well as exhaustive events, as \[A\cap A'=\{\,\,\,\}\]and** \[A\cup A'=S\]**

**(b)** A - B denotes the occurrence of event A but not B. Thus, A - B occurs \[\Leftrightarrow A\]occurs and B does not occur.

Clearly, \[A-B=A\cap B',B-A=B\cap A'\].

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**Addition theorem of probability:**If A and B are any two events in a sample space S, then the probability of occurrence of at least one of the events A and B is given by\[P(A\cup B)=P(A)+P(B)\].

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**Note:**

(i) If A and B are mutually exclusive events, then \[A\cap B=\phi \]and hence, \[P(A\cap B)=0\].

\[\therefore \] \[P(A\cup B)=P(A)+P(B)\]

(ii) Two events A and B are mutually exclusive if and only if** \[P(A\cup B)=P(A)+P(B)\]**

*play_arrow*Introduction*play_arrow*Random Experiments*play_arrow*Events*play_arrow*Probability*play_arrow*Probability*play_arrow*Probability

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