There are four fair dice D_1, D_2, D_3 and D_ | vedclass

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(A) Let $X_1, X_2, X_3$ and $X_4$ be the outcomes of the four dice $D_1, D_2, D_3$ and $D_4$ respectively.Each $X_i \in \{1, 2, 3, 4, 5, 6\}$.The tota

Vedclass · Accepted Answer

$91$

Vedclass · Answer

(A) Let $X_1, X_2, X_3$ and $X_4$ be the outcomes of the four dice $D_1, D_2, D_3$ and $D_4$ respectively.Each $X_i \in \{1, 2, 3, 4, 5, 6\}$.The total number of outcomes is $6^4 = 1296$.We want to find the probability that $X_4 \in \{X_1, X_2, X_3\}$.It is easier to calculate the complement: the probability that $X_4 
otin \{X_1, X_2, X_3\}$.For a fixed value of $X_4 = k$,the number of ways such that $X_1, X_2, X_3 
eq k$ is $5 	imes 5 	imes 5 = 125$.Since there are $6$ possible values for $X_4$,the total number of favorable outcomes for the complement is $6 	imes 125 = 750$.The probability of the complement is $P(X_4 
otin \{X_1, X_2, X_3\}) = \frac{750}{1296} = \frac{125}{216}$.Therefore,the required probability is $1 - \frac{125}{216} = \frac{91}{216}$.

column-$I$	column-$II$
$(A)$ $A$ line from the origin meets the lines $\frac{x-2}{1}=\frac{y-1}{-2}=\frac{z+1}{1}$ and $\frac{x-\frac{8}{3}}{2}=\frac{y+3}{-1}=\frac{z-1}{1}$ at $P$ and $Q$ respectively. If length $PQ=d$,then $d^2$ is	$(p)$ $-4$
$(B)$ The values of $x$ satisfying $\tan ^{-1}(x+3)-\tan ^{-1}(x-3)=\sin ^{-1}\left(\frac{3}{5}\right)$ are	$(q)$ $0$
$(C)$ Non-zero vectors $\vec{a}, \vec{b}$ and $\vec{c}$ satisfy $\vec{a} \cdot \vec{b}=0$,$(\vec{b}-\vec{a}) \cdot(\vec{b}+\vec{c})=0$ and $2\|\vec{b}+\vec{c}\|=\|\vec{b}-\vec{a}\|$. If $\vec{a}=\mu \vec{b}+4 \vec{c}$,then the possible values of $\mu$ are	$(r)$ $4$
$(D)$ Let $f$ be the function on $[-\pi, \pi]$ given by $f(0)=9$ and $f(x)=\frac{\sin \left(\frac{9 x}{2}\right)}{\sin \left(\frac{x}{2}\right)}$ for $x \neq 0$. The value of $\frac{2}{\pi} \int_{-\pi}^\pi f(x) dx$ is	$(s)$ $5$
	$(t)$ $6$

There are four fair dice $D_1, D_2, D_3$ and $D_4$. Each has six faces numbered $1, 2, 3, 4, 5$ and $6$. They are rolled one by one. What is the probability that the number shown on $D_4$ is equal to at least one of the numbers shown on $D_1, D_2$ and $D_3$ (in $/216$)?

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