The random variable $H~B(192,p)$, and $E\left(H\right)$ is $24$ times the standard deviation of $H$. Calculate the value of $p$ and find the value of $k$, given that $P(H=2)=k\times 2^{-379}$.

It is estimated that $1.3\%$ of the matches produced at a factory are damaged in some way. A household box contains $462$ matches. Calculate the expected number of damaged matches in a household box. 

It is estimated that $1.3\%$ of the matches produced at a factory are damaged in some way. A household box contains $462$ matches. Find the variance of the number of damaged matches and the variance of the number of undamaged matches in a household box.[Write your answer correcting to three decimal places]

It is estimated that $1.3\%$ of the matches produced at a factory are damaged in some way. A household box contains $462$ matches. Show that approximately $10.4\%$ of the household boxes are expected to contain exactly eight damaged matches.

It is estimated that $1.3\%$ of the matches produced at a factory are damaged in some way. A household box contains $462$ matches. Calculate the probability that at least one from a sample of two household boxes contains exactly eight damaged matches.  [Write your answer correcting to three decimal places]

On average, $8\%$ of the candidates sitting an examination are awarded a merit. Groups of $50$ candidates are selected at random. How many candidates in each group are not expected to be awarded a merit $?$

On average, $8\%$ of the candidates sitting an examination are awarded a merit. Groups of $50$ candidates are selected at random. Calculate the variance of the number of merits in the groups of $50$.

On average, $8\%$ of the candidates sitting an examination are awarded a merit. Groups of $50$ candidates are selected at random. Find the probability that three, four or five candidates in a group of $50$ are awarded merits.  [Write your answer correcting to three decimal places]

On average, $8\%$ of the candidates sitting an examination are awarded a merit. Groups of $50$ candidates are selected at random. Find the probability that three, four or five candidates in both of two groups of $50$ are awarded merits.  [Write your answer correcting to three decimal places]