A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.

1. Elementary event:

If a random experiment is performed, then each of its outcomes is known as an elementary event.

2. Sample space of an experiment:

The set of all possible outcomes of a random experiment is called the sample space associated with it and it is generally denoted by $S$

3. Simple event and Compound event:

(i) A subset of the sample space associated to a random experiment is said to define a simple event if it has only one outcome.

(ii) A subset of the sample space associated to a random experiment is said to define a compound event if it has more than one outcome.

4. Favourable elementary events:

Let $S$ be the sample space associated with a random experiment and $A$ be an event associated to the experiment. Then elementary events belonging to $A$ are known as favourable elementary events to the event $A$.

5. Probability of an event:

If there are $n$ elementary events associated with a random experiment and $m$ of them are favourable to an event $A$, then the probability of happening or occurrence of $A$ is denoted by $P\left(A\right)$ and is defined as the ratio mn.

6. Relationship between the probability of an event and its complement:

The sum of probability of an event and its complement is $1$. So, given an event $A$, we have $P\left(\overline{A}\right)+P\left(A\right)=1$.

7. Odds ratio: 

Number outcomes in the favour of event $A$ be $m$ and number of outcomes against the event $A$ be $n$, then

(i) The odds in favour of occurrence of the event $A$ are defined by $m:n$ i.e., $P\left(A\right):P\left(\overline{A}\right)$

(ii) The odds against the occurrence of $A$ are defined by $n:m$ i.e., $P\left(\overline{A}\right):P\left(A\right)$.

8. Types of events and the relationship between their probabilities:

(i) Mutually exclusive events:

(a) Two or more events associated to a random experiment are mutually exclusive if they have no common outcomes.

(b) If two events $A$ and $B$ are mutually exclusive, then $P\left(A\cap B\right)=0.$ Similarly, if $A$, $B$ and $C$ are mutually exclusive events, then $P\left(A\cap B\cap C\right)=0.$

(c) All elementary events associated to a random experiment are mutually exclusive.

(ii) Exhaustive events:

(a) Two or more events associated to a random experiment are exhaustive if their union is the sample space. i.e. events $A_1,A_2,...,A_n$ associated to a random experiment with sample space $S$ are exhaustive if $A_1\cup A_2 ...\cup A_n=S.$

(b) All elementary events associated to a random experiment form a system of exhaustive events.

(c) For any event $A$ associated to a random experiment, $A$ and $\overline{A}$ form a pair of exhaustive and mutually exclusive events.

(iii) Independent events:

Two events $A$ and $B$ associated to a random experiment are independent if the probability of occurrence or non-occurrence of $A$ is not affected by the occurrence or non-occurrence of $B$.

NOTE: If $A$ and $B$ are two mutually exclusive events associated to a random experiment, then the occurrence of any one of these two prevents the occurrence of the other i.e., If $A$ occurs, then $P\left(B\right)=0$ and if $B$ occurs, then $P\left(A\right)=0.$ It follows from this that mutually exclusive events associated to a random experiment are not independent and vice-versa.

9. Theorems of probability:

(i) Addition theorem of probability for two events:

If $A$ and $B$ are two events associated with a random experiment, then $P\left(A\cup B\right)=P\left(A\right)+P\left(B\right)-P\left(A\cap B\right)$

(ii) Addition theorem of probability for two mutually exclusive events: 

If $A$ and $B$ are mutually exclusive events, then $P\left(A\cap B\right)=0$ and hence,$P\left(A\cup B\right)=P\left(A\right) +P\left(B\right)$

(iii) Addition theorem of probability for three events:

If $A$, $B$, $C$ are three events associated with a random experiment, then $P\left(A\cup B\cup C\right)=P\left(A\right)+P\left(B\right)+P\left(C\right)-P\left(A\cap B\right)-P\left(B\cap C\right)-P\left(A\cap C\right)+P\left(A\cap B\cap C\right)$

(iv) Probability of occurrence of event $A$ but not $B$:

Let $A$ and $B$ be the two events associated to a random experiment. Then,

(a) $P(\overline{A}\cap B)=P\left(B\right)-P\left(A\cap B\right)$

(b) $P(A\cap \overline{B})=P\left(A\right)-P\left(A\cap \mathrm{B}\right)$

(v) Probability of occurrence of exactly one of the two event:

$(A\cap \overline{B})\cup (\overline{A}\cap B)$ represents the occurrence of exactly one of two events $A$ and $B$. Its probability is given by    

(a) $P\left(\left(A\cap \overline{B}\right)\cup \left(\overline{A}\cap B\right)\right)=P\left(A\right)+P\left(B\right)-2P\left(A\cap B\right)$

(b) $P\left(A\right)+P\left(B\right)-2P\left(A\cup B\right)=P\left(A\cup B\right)-P\left(A\cap B\right).$

10. If $A, B, C$ are three events, then

(i) $P$(At least two of $A, B, C$ occur) $=P\left(A\cap B\right)+P\left(B\cap C\right)+P\left(C\cap A\right)-2P\left(A\cap B\cap C\right)$

(ii) $P$(Exactly two of $A, B, C$ occur) $=P(A\cap B)+P\left(B\cap C\right)+P\left(A\cap C\right)-3P\left(A\cap B\cap C\right)$

(iii) $P$ (Exactly one of $A, B, C$ occurs) $=P\left(A\right)+P\left(B\right)+P\left(C\right)-2P\left(A\cap B\right)-2P\left(B\cap C\right)-2P\left(A\cap C\right)+3P\left(A\cap B\cap C\right).$

11. Conditional probability: 

Let $A$ and $B$ be two events associated with a random experiment. Then, the probability of occurrence of event $A$ under the condition that $B$ has already occurred and $P\left(B\right)\ne 0,$ is called the conditional probability and it is denoted by $P\left(\frac{A}{B}\right)$

12. Conditional probability of independent events:

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

13. Probability of intersection of two events:

If $A$ and $B$ are two events associated with a random experiment, then $P(A\cap B)=P\left(A\right)P\left(\frac{B}{A}\right),$ if $P\left(A\right)\ne 0$ or, $P\left(A\cap B\right)=P\left(B\right)P\left(\frac{A}{B}\right),$ if $P\left(B\right)\ne 0$

14. Probability of intersection of $n$ events: If $A_1, A_2, ..., A_n$ are $n$ events associated with a random experiment, then

$P\left(A_1\cap A_2\cap A_3 ....\cap A_n\right)=P\left(A_1\right)P\left(\frac{A_2}{A_1}\right)P\left(\frac{A_3}{A_1\cap A_2}\right)$…….$P\left(\frac{A_n}{A_1\cap A_2\cap ....\cap A_{n-1}}\right)$

15. Properties of independent events:

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

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

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

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

16. Total Probability Theorem: 

Let $S$ be the sample space and let $E_1, E_2, ..., E_n$ be $n$ mutually exclusive and exhaustive events associated with a random experiment. If $A$ is any event which occurs with $E_1$ or $E_2$ or ... or $E_n,$ then

$P\left(A\right)=P\left(E_1\right)P\left(\frac{A}{E_1}\right)+P\left(E_2\right)P\left(\frac{A}{E_2}\right)+ ... + \dots \dots \dots + P\left(E_n\right)P\left(\frac{A}{E_n}\right)$ or $P\left(A\right)=\sum _{r=1}^{n}P\left(E_r\right)P\left(\frac{A}{Er}\right)$

17. Baye’s theorem: 

Let $S$ be the sample space and let $E_1, E_2..., E_n$ be $n$ mutually exclusive and exhaustive events associated with a random experiment. If $A$ is any event which occurs with $E_1$ or $E_2$ or ... or $E_n,$ then

$P\left(\frac{E_i}{A}\right)=\frac{P\left(E_i\right)P\left({\displaystyle \frac{A}{E_i}}\right)}{{\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}}P\left(E_i\right)P\left(\frac{A}{E_i}\right)}, i=1, 2, \dots , n$

18. Random variable:

Let $S$ be the sample space associated with a given random experiment. Then, a real valued function $X$ which assigns each event $w \in S$ to a unique real number $X \left(w\right)$ is called a random variable.

19. Probability distribution of a random variable:

If a random variable $X$ takes values $x_1, x_2, ...., x_n$ with respective probabilities $p_1, p_2,.... , p_n$ then

 where $p_i\ge 0$ and $\sum _{}{p}_i=1$ is known as the probability distribution of $X$.

20. If $X$ is a random variable with the probability distribution

then,

(i) $P(X\le x_i)=P\left(X=x_1\right)+P\left(X=x_2\right)+ \dots + P(X=x_i)=p_1+p_2 + ....+ p_i$

(ii) $P(X<x_i)=P\left(X=x_1\right)+P\left(X=x_2\right)+ \dots + P\left(X=x_{i-1}\right) = p_1+p_2 + ....+ p_{i-1}$

(iii) $P\left(X\ge x_i\right)=P\left(X=x_i\right)+P\left(X=x_{i+1}\right)+ \dots + P\left(X=x_n\right) = p_i+p_{i+1} + .... + p_n$

(iv) $P\left(X>x_i\right)=P\left(X=x_{i+1}\right)+P\left(X=x_{i+2}\right)+ \dots + P\left(X=x_n\right) = p_{i+1}+p_{i+2} + .... + p_n$

(v) $P\left(X>x_i\right)=1 –P\left(X\le x_i\right)$

(vi) $P\left(X<x_i\right)=1-P\left(X\ge x_i\right)$

(vii) $P\left( x_i\le X\le x_j\right)=P\left(X=x_i\right)+P(X=x_{i+1}) + ... + P\left(X=x_j\right)$

(viii) $P( x_i<X<x_j)=P(X=x_{i+1})+P\left(X= x_{i+2}\right) + \dots + P\left(X=x_{j-1}\right)$

21. Mean of a discrete random variable:

If $X$ is a discrete random variable which assumes values $x_1, x_2, x_3, ...., x_n$ with respective probabilities $p_1, p_2, p_3, ...., p_n,$ then the mean $\overline{X}$ of $X$ is defined as $\overline{X}=p_1{x}_2+p_2{x}_2+p_3{x}_3+....+p_n{x}_n$ or, $\overline{X}=E\left(X\right)=\sum _{i=1}^n{p}_i\mathrm{ }{x}_i$

22. Variance of a discrete random variable:

If $X$ is a discrete random variable which assumes values $x_1, x_2, x_3, ...., x_n$ with the respective probabilities $p_1, p_2, p_3, ...., p_n,$ then variance of $X$ is defined as $\mathrm{Var} \left(X\right)=p_1{\left(x_1-\overline{X}\right)}^2+p_2 {\left(x_2 –\overline{X}\right)}^2+ \dots . + p_n {\left(x_n – \overline{X}\right)}^2$

$\Rightarrow \mathrm{ }\mathrm{Var}\left(X\right)=\sum _{i=1}^n{p}_{\mathrm{ }i}{\left(x_i-\overline{X}\right)}^2,$ where $\stackrel{-}{X}=\sum _{i=1}^n{p}_i{x}_i$ is the mean of $X.$

Also, $\mathrm{Var}\left(X\right)=\sum _{i=1}^n{p}_i\mathrm{ }{x}_i^2-{\left(\sum _{i=1}^n{p}_i{x}_i\right)}^2$

23. Probability distribution function of a Binomial variate:

A random variable $X$ which takes values $0,1,2,...,n$, is said to follow binomial distribution if its probability distribution function is given by$P\left(X=r\right)={}_{\mathrm{ }}{}^{n}C_r{p}^r{q}^{n-r},r = 0,1,2,...,n,$ where $p, q> 0$ such that $p + q=1.$

24. Mean and Variance of a Binomial variate: 

The mean and variance of a binomial variate with parameters $n$ and $p$ are $np$ and $npq$ respectively.