Question

a die is rolled 6 times. let x denote the number of 2s that appear on the die. show that x is a binomial

Answer 1

A die has 6 sides

The probability of getting a 2 in a single throw is $\frac{1}{2}$

The probability of not getting a 2 in a single throw is $\frac{5}{6}$

The probability of getting one 2 in any of the 6 tosses is

i.e X=1 is given by ₆C₁

multiplied by the probability of getting one 2 in a toss which is 1/6 and the probability of not getting a 2 in one toss which is

$\frac{5}{6} .\frac{5}{6} .\frac{5}{6} .\frac{5}{6} .\frac{5}{6}$

so this would combine to ₆C₁ . $[\frac{1}{6}]^{1} .[\frac{5}{6}]^{5}$

Now similarly for X=2

That is the probability of getting two heads in 6 times is

P(X)= ₆C₂ . $[\frac{1}{6}]^{2} .[\frac{5}{6}]^{4}$

(there are two chances of getting 1/6 and 4 chances of getting 5/6)

Similarly for X=3

P(X)= ₆C₃ . $[\frac{1}{6}]^{3} .[\frac{5}{6}]^{3}$

and so on.

Hence this will sum to

P(X=a)= nCa X$p^{a}$ X $x q^{n-a}$

which is a binomial distribution where

N is the number of tosses

a is the number of desired results or successes

P is the probability of success

q is the probability of failures

Hence this situation follows binomial distribution