Question

A dice is rolled n times. If the probability of at least one 6 is to be greater or
equal to 1/2, find n.​

Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

A dice is rolled 'n' times.

Let p represents the probability of getting 6

and q represents the probability of not getting 6.

Probability of getting 6 is given by,

$\rm :\longmapsto\:p = \dfrac{1}{6}$

and

Probability of not getting 6 is given by

$\rm :\longmapsto\:q = \dfrac{5}{6}$

We know,

Binomial Distribution consisting of n independent trials is given by

$\boxed{ \bf{P(r)\: = \: ^nC_r \: {p}^{r} \: {q}^{n - r} }}$

where,

• n is number of independent trials

• p is probability of success

• q is probability of failure

• r is random variable

It is given that

$\rm :\longmapsto\:P(getting \: atleast \: 1 \: six) \geqslant \dfrac{1}{2}$

$\rm :\longmapsto\:P(1) + P(2) + P(3) + - - - + P(n) \geqslant \dfrac{1}{2}$

$\rm :\longmapsto\:1 - P(0) \geqslant \dfrac{1}{2}$

$\rm :\longmapsto\: - P(0) \geqslant \dfrac{1}{2} - 1$

$\rm :\longmapsto\: - P(0) \geqslant \dfrac{1 - 2}{2}$

$\rm :\longmapsto\: - P(0) \geqslant \dfrac{ - 1}{2}$

$\rm :\longmapsto\:P(0) \leqslant \dfrac{1}{2}$

$\rm :\longmapsto\: ^nC_0 \: {p}^{0} {q}^{n} \: \leqslant \dfrac{1}{2}$

$\rm :\longmapsto\:1 \times 1 \times {\bigg(\dfrac{5}{6} \bigg) }^{n} \leqslant \dfrac{1}{2}$

$\rm :\longmapsto\: {\bigg(\dfrac{5}{6} \bigg) }^{n} \leqslant \dfrac{1}{2}$

$\rm :\longmapsto\: {\bigg(\dfrac{6}{5} \bigg) }^{n} \geqslant 2$

$\rm :\longmapsto\: {6}^{n} \geqslant 2 \times {5}^{n}$

Now, using hit and trial method

For n = 1

$\rm :\longmapsto\:6 \geqslant 10$

not possible.

For n = 2

$\rm :\longmapsto\:36 \geqslant 50$

not possible.

For n = 3

$\rm :\longmapsto\:216 \geqslant 250$

not possible.

For n = 4

$\rm :\longmapsto\:1296 \geqslant 1250$

which is true.

$\bf\implies \:n \geqslant 4$

Additional Information :-

In Binomial Distribution,

• Mean of Binomial Distribution = np

• Variance of Binomial Distribution = npq

• Mean > Variance

where,

• n is number of independent trials

• p is probability of success

• q is probability of failure

• r is random variable