Exercise

Two dice are rolled simultaneously. Find the probability that the sum of the numbers on the faces is neither divisible by $3$ nor by $4$.

The standard deviation of $20$ observations is $\sqrt{5}$. If each observation is multiplied by $2$, find the standard deviation and variance of the resulting observations.

The diameters of circles (in mm) drawn in a design are given below, calculate variance [Use short-cut method]

$\begin{array}{|ll|}\hline \text{ Diameter }& \text{ Number of Circles }\\ 33-36& 10\\ 37-40& 20\\ 41-44& 30\\ 45-48& 40\\ 49-52& 50\\ \hline\end{array}$

The diameters of circles (in mm) drawn in a design are given below, calculate variance [Use short-cut method]

$\begin{array}{|ll|}\hline \text{ Diameter }& \text{ Number of Circles }\\ 33-36& 15\\ 37-40& 17\\ 41-44& 21\\ 45-48& 22\\ 49-52& 25\\ \hline\end{array}$

$A, B, C$ are three events which are equally likely but not forming an exhaustive system of events. Then, show that $P\left(A\right)=P\left(B\right)=P\left(C\right)\ne \frac{1}{3}$. Give one example of such a situation.

Given that $\mathrm{P}\left(\mathrm{A}\right)=0.35, \mathrm{P}\left(\mathrm{B}\right)=0.73$ and $\mathrm{P}(\mathrm{A}\cap \mathrm{B})=0.14,$find $\mathrm{P}(\overline{\mathrm{A}}\cup \overline{\hspace{0.17em}\mathrm{B}})$

State and prove addition theorem on probability.

The diameters of circles (in mm) drawn in a design are given below, calculate standard deviation.  [Use short-cut method]

$\begin{array}{|ll|}\hline \text{ Diameter }& \text{ Number of Circles }\\ 33-36& 15\\ 37-40& 17\\ 41-44& 21\\ 45-48& 22\\ 49-52& 25\\ \hline\end{array}$