Class 12 Maths Probability Binomial Distribution

Binomial Distribution:

A random variable X taking values 0, 1, 2, ..., n is said to have a binomial distribution
with parameters n and p, if its probability distribution is given by

P (X = r) = ncr pr qn–r,
where q = 1 – p and r = 0, 1, 2, ..., n.

Example: If a coin is tossed 10 times, find the probability of exactly 5 heads.

Solution: let X denote the number of heads.
So X has the binomial distribution with n = 10 an p = ½
∴ P(X = x) = nCx qn-x px , x = 1,2,3....,n
∴ P(X = x) = 10cx(1/2)10-x(1/2)x = 10cx(1/2)10.
so P(X = 6) = 10c5(1/2)10= (10 !)/ (5 !×5!) ×( 1/210) = 63/ 256

Example:A bag consists of 10 balls each marked with one of the digits 0 to 9. If four balls are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?

Solution:
Let X denote the number of balls marked with the digit 0 among the 4 balls drawn.
Since the balls are drawn with replacement, the trials are Bernoulli trials.
X has a binomial distribution with n = 4 and P= 1/10

∴ Q= 1-P =1− 1/10 = 9/10

∴ P(X = x) = nCx qn-x px , x = 1,2,3....,n
= 4Cx ( 9/10 )n−x( 1/10 )x

P (none marked with 0) = P (X = 0)
= 4C0 ( 9/10 )4 ( 1/10 )0
= 1×( 9/10)4

= ( 9/10 )4

Example: Find the probability of getting 5 exactly twice in 7 throws of a dice.

Solution:
The repeated tossing of a dice are Bernoulli trials. Let X represent the number of times of getting 5 in 7 throws of the dice.
Probability of getting 5 in a single throw of the dice is P=16
∴Q=1-P= 1− 1/6 = 5/6

Clearly, X has the probability distribution with n = 7 and P=16
∴P(X = x) = nCx qn-x px
= 7Cx (5/6)7−x (1/6)x

P (getting 5 exactly twice) = P(X = 2)
=7C2 (5/6)5 (1/6)2
= 21×(5/6)5× 1/36
= 7 /12 ×(5/6)5

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