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) = ^{n}c_{r} p^{r} q^{n–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) =

∴ P(X = x) =

so P(X = 6) =

* 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) = ^{n}C_{x} q^{n-x} p^{x} , x = 1,2,3....,n

= ^{4}C_{x} ( 9/10 )^{n−x}( 1/10 )^{x}

P (none marked with 0) = P (X = 0)

= ^{4}C_{0} ( 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) = ^{n}C_{x} q^{n-x} p^{x}

= ^{7}C_{x} (5/6)^{7−x} (1/6)^{x}

P (getting 5 exactly twice) = P(X = 2)

=^{7}C_{2} (5/6)^{5} (1/6)^{2}

= 21×(5/6)^{5}× 1/36

= 7 /12 ×(5/6)^{5}

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