3 Marks Question

Question 13 Marks

If $E_1, E_2, E_3$ are three mutually exclusive events and exhaustive events of an experiment such that-
$2 P\left(E_1\right)=3 P\left(E_2\right)=P\left(E_3\right)$, then find $P\left(E_1\right)$.

Answer

Since, $E_1, E_2, E_3$ are mutually exclusive and exhaustive events, so $E_1 \cap E_2=\phi, E_2 \cap E_3=\phi$, $E_1 \cap E_3=\phi, E_1 \cap E_2 \cap E_3=\phi$ and $E_1 \cup E_2 \cup E_3=S 1$ $\therefore \quad P\left(E_1 \cup E_2 \cup E_3\right)=P\left(E_1\right)+P\left(E_2\right)+P\left(E_3\right)$$-P\left(E_1 \cap E_2\right)-P\left(E_2 \cap E_3\right)$$-P\left(E_1 \cap E_3\right)+P\left(E_1 \cap E_2 \cap E_3\right)$
$\begin{array}{l}=P\left(E_1\right)+\frac{2}{3} P\left(E_1\right)+2 P\left(E_1\right) \\ =\frac{11}{3} P\left(E_1\right)\end{array}$
$\Rightarrow \quad P(S)=\frac{11}{3} P\left(E_1\right)$
$\Rightarrow \quad 1=\frac{11}{3} P\left(E_1\right)$
$\Rightarrow \quad P\left(E_1\right)=\frac{3}{11}$.

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Question 23 Marks

Three identical dice are rolled. Find the probability that the same number appears on each of them.

Answer

Since three identical dice are rolled, so number of elements in the sample space $S$ is
$n(S)=6^3=216$
Let E be the event of getting same number on each of them i.e., $E=\{(1,1,1),(2,2,2),(3,3,3),(4,4,4)$, $(5,5,5),(6,6,6)\}$
$\begin{array}{ll}\Rightarrow & n(E)=6 \\ \therefore & P(E)=\frac{n(E)}{n(S)}=\frac{6}{216}=\frac{1}{36} .\end{array}$
$\therefore$ The probability that the same number appears on all the three dice is $1 / 36$.

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Applied Maths 3 Marks Question

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