Probability

The theory of probability originated in the middle of seventeenth century.

Fermat (1601-1665), Pascal (1623-1662) Hugger (1629-1695) James Bernoulli: (1663)

Experiment : An experiment is observing something happen or conducting something which results in an outcome.

Deterministic Expt $\rightarrow$ If after conducting the expt we know the outcome of the expt.

$\mathrm{H}_2 \mathrm{O} \rightarrow \text { water }$

Non-deterministic or Random Experiments When an expt is conducted but the outcome cannot be predicted in advance, then it is called a random expt.

Long term behaviour - Statistical regularity,

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

We usually use the notation $S$ or $\Omega$ to denote the sample space.

Suppose too coins are tossed.

$S={(H, H),(H, T),(T, H),(T, T)} .$

2. Suppose a coin and a die is tossed together.

S = ${(H, 1),(H, 2), (H, 6),(T, 1),(T, 6)}$

3. Time of the day

S=(m, n, p)

$m=1,2, \ldots, 12,$

$n=0,1, \ldots, 54$

$p=0,1, \ldots 59 .$

If we consider continuous time, then we may write the sample space as

$\Omega=(0,24)$

4. Suppose we consider a $100 \mathrm{nt}$ sprint race, say' of olympic standard

$P_1, \ldots, p_8$

If we record the winner, then the sample space is $\quad S_1={P_1, \cdots, P_8}$

If we are interested in the time $f$ the winner then the sample space can be

$S_2=(9: 50,10: 00) \quad \text { (in seconds) }$

5. Suppose we record the number of accidents taking place in a city in an year.

$S={0, 1,2, \ldots \ldots .}$

life of an organism:

$\begin{aligned} & {[0,100)} \ & {[0,150)} \end{aligned}$

Amount of rainfall during a monsoon season $S=[0,200)$ (in cm)

$E=[50,75] \rightarrow$ the amount of rainfall is between $50$ to $75 \mathrm{~cm}$.

$E \subset S$

Sure event: $\rightarrow$ the event that will certainly happen

$\rightarrow$ We use $S$ to denote the sure event

Impossible event:

Let $A$ and $B$ be tor events

$A \cup B \rightarrow$ Occurrence of either $A$ or $B$ or (both occurence of atleast one of A and B)

$\bigcup_{i=1}^n A_i=A_1 \cup A_2 \cup \cdots \cup A_n$

$=\text { occurrence of at leas one } A_i$

$(i=1, \cdots n)$

$\bigcup_{i=1}^{\infty} A_i=$ occurrence of at least one $A_i$

$(i=1,2, \ldots)$

$A \cap B \rightarrow$ simultaneous occurrence of both $A$ and $B$

$\cap_{i=1}^n A_i=A_1 \cap A_2 \cap \cdots \cap A_n$

$\rightarrow$ simultaneous occurrence of $A_1, \ldots, A_n$

$\bigcap_{i=1}^{\infty} A_i$

$\rightarrow$ simultaneous occurrence of $A_1, A_2, \ldots$.

$A-B=A \cap B^c$

$\rightarrow$ Occurrence of A but not of B

$\rightarrow$ A occur but B does not occur

$A^c \rightarrow \text { not occurrence of } A$

$A \cap B=\phi$ $\rightarrow$ disjoint or mutually exclusive events.