Probability

Some Important Points about Trial, Coin, Dice, Playing cards and Events

Coin. A coin has two sides, Head and Tail. If an event consists of more than one coin, coins are regarded as distinct if not otherwise stated.

Dice. A die (cubical) has six faces maked 1, 2, 3, 4, 5, 6. As in the case of coins, if we have more than one die, all dice are regarded as distinct if not otherwise stated.

Playing cards. A pack of playing cards has 52 cards. There are 4 suits (spade, heart, diamond and club) each having 13 cards. There are two colours red (heart and diamond) and black (spade and club) each having 26 cards.
In thirteen cards of each suit, there are 3 face cards namely king, queen and jack so there are in all 12 faces cards (4 king, 4queens and 4 jacks)
Also there are 16 honour cards, 4 of each suit namely ace, king queen and jack.
Trial, Events. Let a random experiment be repeated under identical conditions. Then the experiment is called a Trial and the possible outcomes of the experiment are called elementary events or cases. e.g, Tossing of a coin is a trial and getting head or tail is an event. Throwing a die is a trial and getting 4 in its upper face is an event.

Some results
1. Probability of an event A i.e. P (A)= $\frac{\mathrm{Number}\mathrm{of}\mathrm{favourable}\mathrm{cases}}{\mathrm{Total}\mathrm{number}\mathrm{of}\mathrm{equally}\mathrm{likely}\mathrm{cases}}$

2. Probability of an impossible event is zero. i.e, P( $\phi$ )= 0

3. Probability of a sure event is one i.e, P(S) = 1 where S is the sure event. Example:- Suppose a die is thrown. The sample space S= { 1,2,3,4,5,6}. Let E= { A +ve integer less than 7}= {1,2,3,4,5,6}. This is a sure event .
Again if E= { A +ve integer greater than 6} = {} = $\phi$. This is an impossible event.

4. Mutually exclusive events

A set of events is said to be mutually exclusive if the happening of one excludes the

happening of the other. Let $E_1$= a sum of 5= {(1,4), (2,3), (3,2), (4,1)} and $E_2$= a sum

of 7 = {(1,6), (2,5), (3,4),(4,3), (5,2), (6,1)}

Thus clearly $E_1$ $\cap$ $E_2$ = $\phi$

$\therefore$ $E_1$, $E_2$ are mutually exclusive.

5. Exhaustive events
A set of events is said to be exhaustive if the performance of the experiment always

results in the occurrence of at least one of them.

6. (i) P (A$\cup$B) = P (A) + P (B) - P (A$\cap$B), where A, B are any two events.

(ii) P (A$\cup$B$\cup$C) = P (A) + P (B) where A and B are mutually exclusive events.

7. (i) 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)

where A, B, C are any events.

(ii) P(A$\cup$B$\cup$C) = P(A) + P (B) + P(C)

if A, B, C are mutually exclusive events.

8. 0 $\le$ P(A) $\le$ 1 for an event A.

9. P($\overline{A}$) = 1 - P(A)

(P($\overline{A}$) = Prob. of not happening the event A)

10. P(A$\cup$ $\overline{A}$) = P(S) = 1

11. P(A$\cap$B) = P(A). P(B) if A, B are independent events.

12. P(A$\cap$B) = P(A) . P(B/A) = P(B) . P(A/B) where P(B/A) = Prob. of B when A has

already occurred.

= conditional probability of B.

13. (a) P (exactly one of A, B occurs)

= P (A$\cup$B) - P(A$\cap$B)

= P($\overline{A}\cup$$\overline{B}$
) - P($\overline{A}\cap$$\overline{B}$
)

(b) P(AB) $\le$ P(A) $\le$ P(A +B) $\le$ P(A) + P(B)

(c) P ($\overline{A}$ $\cap$ B) = P(B) - P(A$\cap$B)

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

(d) If B $\underset{\_}{\subset }$ A, then

(i) P(A$\cap$ $\overline{B}$) = P(A) - P(B)

(ii) P(B)$\underset{\_}{\subset }$ P(A)

P(B/A) = $\frac{P(A\cap B)}{P(A)}$ where A $\ne$ 0

P(A/B) = $\frac{P(A\cap B)}{P(B)}$ where B $\ne$ 0

e) If p = prob. of the happening of an event. q= prob. for not happening the same event

then p+q= 1. Also 0 $\le$ p $\le$ 1.

14. Baye's Theorem

P($E_i/A)$ =$\frac{P(E_1)\mathrm{.}P(A/E_1)}{\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}P(E_i)\mathrm{.}P(A/E_i)}$

where $E_1$, $E_2$, ..........$E_n$ are mutually exclusive events.

15. Mathematical Expectation
Let x be discrete random variable which assume the valves $x_1$, $x_2$,........$x_n$ with the

corresponding probabilities $p_1$, $p_2$,......$p_n$.

Then the expected value of X ie. E(x) is defined as

E(X)= $\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}$ $x_i{p}_i$ where $\underset{i=1}{\overset{n}{\mathrm{\Sigma }}}$ $p_i$= 1

i.e. E(X) = $x_1{p}_1$ + $x_2{p}_2$+........+$x_n{p}_n$

where $p_1+p_2$+ ..... $p_n$= 1

16. The odds in favour of an event is given by P(A) / P ($\overline{A}$)

The odds against the happening of an event A is given by P ($\overline{A}$)/P(A).

17. If A, B are mutually exclusive and exhaustive events, then A$\cup$B= S

$\therefore$ P(A) + P(B) = P(A$\cup$B)= P(S)= 1

ie. P(A) + P(B) = 1

18. If A, B are mutually exclusive events, then A$\cap$B = $\phi$ $\Rightarrow$ P(A$\cap$B) = P($\phi$) = 0 $\Rightarrow$ P(A$\cap$B) = 0

19. Binomial Distribution

A random variable X is said to follow Binomial Distribution if it takes only non-

negative values and its probability function is given by

P(X=r) = P(r) = ${}^n{C}_r{p}^r{q}^{n-r},r$= 0, 1, 2, 3,......... and q = 1 - p and = 0 otherwise

Here n and p are called parameters of the Binomial Distribution.

20. Properties of Binomial Distribution

(i) Mean of Binomial Distribution = np

variance ${\sigma }^2$ = npq, S.D= $\sqrt{\mathrm{npq}}$

(ii) Since 0$<$q$<$1, $\therefore$ variance = npq $<$ np. Hence for Binomial Distribution, variance

is less than Mean.

(iii) Mode of the Binomial Distribution is that value of the variable which occurs with

the largest probability.

It may have either one mode or two modes. If (n+1) p =k, is an integer, then k and

(k-1) are the two modes. If (n+1) p=k is not an integer, then mode= [(n+1)p], where [x]

is a greatest integer function.

(iv) $\underset{r=0}{\overset{n}{\mathrm{\Sigma }}}$P(r) = ${(q+p)}^n$=1

(v) If two independent random variables X and Y follow Binomial Distribution with

parameters $n_1$, $p_1$ and $n_2$, $p_2$ respectively, then their sum (X+Y) also follows

binomial distribution with parameters $n_1$, $n_2$, p.

(vi) If n independent trials are repeated N- times, N sets of trials are obtained. The

expected frequency of r successes = N .${}^n{C}_r{p}^r{q}^{n-r}\mathrm{.}$

Thus the expected frequency of 0, 1, 2, 3,.....n successes are successive terms of

Binomial Expansion of N${(q+p)}^n$.

(vii) P(r+1) = m$\left[\frac{n-r}{r+1}\mathrm{.}\frac{p}{q}\right]$ P(r) is called Recurrence formula for Binomial

Distribution.

21. Poisson Distribution

(As a limiting case of Binomial Distribution)

Poisson Distribution was derived in 1837 by a French Mathematician Simeon D

Poisson (1781 - 1840). This distribution can be regarded as a limiting case of Binomial

Distribution under the following conditions.

(a) n, the number of trials is very large i.e N $\to$ $\infty$

(b) p $\to$ 0 ie, prob. for success for each trial $\to$ 0

(c) np = $\lambda$ (say) is finite.

(d) P(X= r) i.e P(r) = $\frac{e^{-\lambda }{\lambda }^r}{r!}$, r = 0, 1, 2..... and $\lambda$ = np (Here $\lambda$ is called parameter of

distribution ) where $\lambda$ is the number of successes.

(e) P( r+1) = $\frac{\lambda }{r+1}$ P(r) is called Recurrence formula

(f) $\underset{r=0}{\overset{\infty }{\mathrm{\Sigma }}}$ P(r) =1

(g) If X and Y are independent variates with parameters ${\lambda }_1$ and ${\lambda }_2$ respectively, then

(X+Y) also follows Poisson distribution with parameter (${\lambda }_1$ + ${\lambda }_2$).

(h) IN Poisson distribution

Mean = variance = $\lambda$

Since Mean = $\begin{array}{c}\mathrm{Lt}\\ n\to \infty \end{array}$(np) = $\begin{array}{c}\mathrm{Lt}\\ n\to \infty \end{array}$($\lambda$) = $\lambda$

[variance = $\begin{array}{c}\mathrm{Lt}\\ n\to \infty \end{array}$(npq) = $\begin{array}{c}\mathrm{Lt}\\ n\to \infty \end{array}$np (1-p)

= $\begin{array}{c}\mathrm{Lt}\\ n\to \infty \end{array}$(np) $(1-\frac{\lambda }{n})$= $\lambda$(1-0) = $\lambda$

22. Normal Distribution

Normal distribution is defined by the probability density function

P (X) = $\frac{1}{\sigma \sqrt{2\pi }}$ $e^{\frac{-{(x-\mu )}^2}{2{\sigma }^2}}$

where -$\infty$ $<$ $\times$ $<$ $\infty$ (Here $\mu$ = Mean, $\sigma$ = variance)

Properties of Normal Distribution

(i) Normal Distribution is a limiting case of Binomial Distribution when the number of

trials n is very large and neither p nor q is small.

(ii) Normal Distribution is a limiting case of Poisson distribution when its mean $\mu$ is

large.

(iii) For a normal distribution

Mean = Mode = Median

(iv) Normal curve is uni-modal

(v) For a normal distribution,

Mean Deviation = $\frac{4}{5}$ of S.D

(vi) Area under normal curve = 1

(vii) Normal curve is symmetrical about mean. It is bell shaped as shown in figure.

(viii) The height of the normal curve is maximum at the mean valve.

(ix) $Q_3$- Median = Median - $Q_1$ for a normal curve.