Notes On Binomial Distribution - CBSE Class 12 Maths

For n Bernoulli trials:

    X

      0

      1

      2

    ...

      n

  P(X)

1st term of (q + p)n

2nd term of (q + p)n

3rd term of (q + p)n

...

(n + 1)th term of (q + p)n

     X

        0

        1

         2

...

          n 

   P(X)

nC0qn-0p0

nC1qn-1p1

nC2qn-2p2

...

nCnqn-npn

This probability distribution of a random variable, x, is called a binomial distribution.

Thus, the general formula for the probability of a given number of successes is given by…

P(X = x) = P(x) = nCxqn-xpx

            = n!/(x!(n-x)!) qn-xpx, where x = 0,1,2,....,n and q = 1 - p

P (x) is known as the probability function of the binomial distribution.

Ex:

Find the probability of getting two heads, if a coin is tossed six times.

Sol:

Let the event of getting a head on tossing a coin be a success, and the event of getting a tail be a failure.

Let X denote the number of heads obtained, when a coin is tossed six times.

Let p be the probability of getting a head and q be the probability of getting a tail, if a coin is tossed once.

p = 1/2 and q = 1/2

Number of trials, n = 6

∴ X has a binomial distribution with n =6 and p = 1/2

We have P(X = x) = nCxqn-xpx , where x = 0,1,2,3,....,n

The probability of getting 2 heads = P(X = 2)

= 6C2(1/2)6-2(1/2)2

= 6C2(1/2)6

= 6!/(2!4!) (1/2)6

= (6x5)/(2x1)  . (1/2)6

= 15/64

Hence, the probability of getting 2 heads, when a coin is tossed six times = 15/64

Summary

For n Bernoulli trials:

    X

      0

      1

      2

    ...

      n

  P(X)

1st term of (q + p)n

2nd term of (q + p)n

3rd term of (q + p)n

...

(n + 1)th term of (q + p)n

     X

        0

        1

         2

...

          n 

   P(X)

nC0qn-0p0

nC1qn-1p1

nC2qn-2p2

...

nCnqn-npn

This probability distribution of a random variable, x, is called a binomial distribution.

Thus, the general formula for the probability of a given number of successes is given by…

P(X = x) = P(x) = nCxqn-xpx

            = n!/(x!(n-x)!) qn-xpx, where x = 0,1,2,....,n and q = 1 - p

P (x) is known as the probability function of the binomial distribution.

Ex:

Find the probability of getting two heads, if a coin is tossed six times.

Sol:

Let the event of getting a head on tossing a coin be a success, and the event of getting a tail be a failure.

Let X denote the number of heads obtained, when a coin is tossed six times.

Let p be the probability of getting a head and q be the probability of getting a tail, if a coin is tossed once.

p = 1/2 and q = 1/2

Number of trials, n = 6

∴ X has a binomial distribution with n =6 and p = 1/2

We have P(X = x) = nCxqn-xpx , where x = 0,1,2,3,....,n

The probability of getting 2 heads = P(X = 2)

= 6C2(1/2)6-2(1/2)2

= 6C2(1/2)6

= 6!/(2!4!) (1/2)6

= (6x5)/(2x1)  . (1/2)6

= 15/64

Hence, the probability of getting 2 heads, when a coin is tossed six times = 15/64

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