Statistics Standard 11>Chapter -1 - Elementary Probability Theory>EXERCISE>Q 26

EASY

11th Tamil Nadu Board

IMPORTANT

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Define exhaustive events.

Important Points to Remember in Chapter -1 - Elementary Probability Theory from Tamil Nadu Board Statistics Standard 11 Solutions

1. Random Experiment:

An experiment whose outcomes cannot be predicted or determined in advance is called a random experiment.

2. Elementary event:

Each outcome of a random experiment is known as an elementary event.

3. Sample Space:

The set of all possible outcomes (elementary events) of a random experiment is called the sample space associated with it.

4. Event:

A subset of the sample space associated with a random experiment is called an event.

5. Outcome:

An event is said to occur if any one of the elementary events belonging to it is an outcome.

6. Types of Events:

(i) Certain or Sure event:

An event associated with a random experiment is called a certain event if it always occurs whenever the experiment is performed. The sample space associated with a random experiment defines a certain event.

(ii) Impossible event:

The null set of the sample space defines an impossible event.

(iii) Simple or elementary event:

Each outcome of a random experiment is called an elementary event.

(iv) Compound events:

If an event has more than one outcome is called compound events.

(v) Complementary event:

Given an event $A$ , the complement of $A$ is the event consisting of all sample space outcomes that do not correspond to the occurrence of $A$ .

7. Mutually Exclusive Events:

Two or more events associated with a random experiment are said to be mutually exclusive or incompatible events if the occurrence of any one of them prevents the occurrence of all others i.e. no two or more of them can occur simultaneously in the same trial. If $A$ and $B$ are mutually exclusive events, then $A \cap B = ϕ$

8. Exhaustive Events:

Events $A_{1}, A_{2}, A_{3}, . ..., A_{n}$ associated with a random experiment with sample space $S$ are exhaustive if $A_{1} \cup A_{2} \cup ..... \cup A_{n} = S$

9. Mutually Exclusive and Exhaustive Events:

Let $S$ be the sample space associated with a random experiment. A set of events $A_{1}, A_{2}, ..., A_{n}$ is said to form a set of mutually exclusive and exhaustive system of events if

(i) $A_{1} \cup A_{2} \cup . ... \cup A_{n} = S$

(ii) $A_{i} \cap A_{j} = ϕ$ for $i \neq j$

10. Probability function:

(i) $0 \leq P (w_{i}) \leq 1$ for all $w_{i} \in S$

(ii) $P (S) = 1$ i.e. $P (w_{1}) + P (w_{2}) + . .... + P (w_{n}) = 1$

(iii) For any event $A \subset S$ , $P (A) = \sum_{W_{k} \in A} P (w_{k})$ , the number $P_{(W_{k})}$ is called probability of elementary event $w_{k}$ .

11. Probability of an event:

(i) $P (A) = \frac{m}{n} = \frac{Favourable number of elementary events}{Total number of elementary events}$

(ii) $P (A^{'}) = 1 - P (A)$ .

12. Addition Rule of Probabilities:

(i) $P (A \cup B) = P (A) + P (B) - P (A \cap B) .$

(ii) If $A$ and $B$ are mutually exclusive events, then $P (A \cup B) = P (A) + P (B) .$

(iii) $P (A \cup B \cup C) = P (A) + P (B) + P (C) - P (A \cap B) - P (B \cap C) - P (C \cap A) + P (A \cap B \cap C)$

(iv) If $A$ and $B$ are two events associated with a random experiment, then

(a) $P (A \cap B) = P (B) - P (A \cap B)$ i.e. probability of occurrence of $B$ only

(b) $P (A \cap B) = P (A) - P (A \cap B)$ i.e. probability of occurrence of $A$ only

(c) Probability of occurrence of exactly one of $A$ and $B$ is $P (A) + P (B) - 2 P (A \cap B) = P (A \cup B) - P (A \cap B)$

13. Conditional probability of independent events:

If A and B are independent events associated with a random experiment, then $P (\frac{A}{B}) = P (A)$ .

14. Probability of intersection of two events:

$P (A \cap B) = P (A) P (\frac{B}{A}),$ if $P (A) \neq 0$ or, $P (A \cap B) = P (B) P (\frac{A}{B}),$ if $P (B) \neq 0$

15. Probability of intersection of n events:

$P (A_{1} \cap A_{2} \cap A_{3} \dots \cap A_{n}) = P (A_{1}) P (\frac{A_{2}}{A_{1}}) P (\frac{A_{3}}{A_{1} \cap A_{2}}) \dots P (\frac{A_{n}}{A_{1} \cap A_{2} \cap \dots \cap A_{n - 1}})$

16. Properties of independent events:

If $A$ and $B$ are independent events associated with a random experiment, then

(i) $\bar{A}$ and $B$ are independent events.

(ii) $A$ and $\bar{B}$ are independent events.

(iii) $\bar{A}$ and $\bar{B}$ are also independent events.

17. Total Probability Theorem:

$P (A) = P (E_{1}) P (\frac{A}{E_{1}}) + P (E_{2}) P (\frac{A}{E_{2}}) + \dots + P (E_{n}) P (\frac{A}{E_{n}})$ or $P (A) = \sum_{r = 1}^{n} P (E_{r}) P (\frac{A}{E_{r}})$ .