Consider a random experiment whose outcomes can be
classified as success or failure. It means that experiment results in only two
outcomes E_{1}(success) or E_{2} (failure). Further assume
that experiment can be repeated several times, probability of success or
failure in any trial are p and q (p + q = 1) and don’t vary from trial to trial
and finally different trials are independent. Such a experiment is called Binomial
experiment and trials are said to be binomial trials. For instance tossing of a
fair coin several times, each time outcome would be either a success (say
occurrence of head) or failure (say occurrence of tail).

A probability distribution representing the binomial trials is said to binomial distribution.

Let us consider a Binomial experiment which has been
repeated ‘n’ times. Let the probability of success and failure in any trial be
p and q respectively and we are interested in the probability of occurrence of
exactly ‘r’ successes in these n trials. Now number of ways of choosing ‘r’
success in ‘n’ trials = ^{n}C_{r}. Probability of ‘r’ successes
and (n-r) failures is p^{r}×q^{n}^{-}^{r}. Thus probability
of having exactly r successes = ^{n}C_{r}×p^{r}×q^{n}^{-}^{r}.

Let ‘X’ be random variable representing the number of successes, then

P(X = r) = ^{n}C_{r}×p^{r}×q^{n}^{-}^{r} (r = 0, 1, 2, L , n)

q Probability of utmost ‘r’ successes in n trials =

q Probability of atleast ‘r’ successes in n trials =

q Probability of
having 1^{st} success at the r^{th} trial = p×q^{r}^{-}^{1}

A die is thrown 7 times. What is the chance that an odd number turns up .

(i) exactly 4 times

(ii) at least 4 times?

Probability of success =

&⇒ p = and

(i) For exactly 4 successes required probability

= ^{7}C_{4}.

(ii) For atleast 4 successes required probability

= ^{7}C_{4}

=

.

A and B play a series of games which cannot be drawn and p, q are their respective chances of winning a single game. What is the chance that A wins m games before B wins n games.

For this to happen, A must win atleast m out of the first m + n - 1 games, \ The required probability

= ^{m + n - 1}C_{m}p^{m}q^{n
- 1} + ^{m + n }^{-}^{ 1}C_{m + 1} p^{m + 1} q^{n
- 2} + .... + ^{m + n }^{-}^{ 1 }C_{m + n – 1}p^{m + n -1}.