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NDA Math Probability

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Question : 43 of 100

Marks: +1, -0

A special dice with numbers

1,-1,2,-2,0

and 3 is thrown thrice. What is the probability that the sum of the numbers occurring on the upper face is zero?

$\frac{1}{72}$
$\frac{1}{8}$
$\frac{7}{72}$
$\frac{25}{216}$

Solution:

Total no. of elementary events

=6^{3}

.

Favourable no. of elementary events

= coefficient of

x^{0}

in

[x+x^{-1}+x^{0}+x^{-2}+x^{2}+x^{3}]^{3}

= coefficient of

x^{0}

in

\left[\;\frac{1+x+x^{2}+x^{3}+x^{4}+x^{5}}{x^{2}}\right]^{3}

= coefficient of

x^{6}

in

[1+x+x^{2}+x^{3}+x^{4}+x^{5}]^{3}

= coefficient of

x^{6}

in

\left[\;\frac{1-x^{6}}{1-x}\right]^{3}

= coefficient of

x^{6}

in

[1-x^{6}]^{3}[1-x]

= coefficient of

x^{6}

in

[1-{}^3C_1 x^{6}+\dots][1-x]^{-3}

= coefficient of

x^{6}

in

(1-x)^{-3}\cdot{}^3C_1

coeff. of

x^{0}

in

(1-x)^{-3}

={}^{6+3-1}C_{3-1}-{}^3C_1

={}^{8}C_{2}-{}^3C_1=\;\frac{8!}{6!\,2!}-\;\frac{3!}{2!}

=\;\frac{8\times7}{2}-3=25

Required probability

=\frac{25}{216}

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