In a large school one person in five is left-handed.

How large must a random sample be if the probability that it contains at least one left-handed person is greater than $0.95?$

In a mass production of batteries, the probability that one battery is defective is $0.5\%$. Given that the batteries are packed randomly in packs of $10$ and quality control inspect batteries from randomly selected packs. Find the probability that a selected pack contains one defective battery.

In a mass production of batteries, the probability that one battery is defective is $0.5\%$. Given that the batteries are packed randomly in packs of $10$ and quality control inspect batteries from randomly selected packs. Find the probability that a selected pack contains not more than one defective battery.

In a mass production of batteries, the probability that one battery is defective is $0.5\%$. Given that the batteries are packed randomly in packs of $10$ and quality control inspect batteries from randomly selected packs. Find the probability that a selected pack contains exactly $1$ defective battery, given that it has not more than one defective battery.

A random variable $X$ follows a binomial distribution with mean $3$ and variance $1.2$. Find the value of the parameters $n$ and $p$ of the distribution of $X$.

A packing machine produces bags of rice whose weights are normally distributed with mean $50.1$ kg and standard deviation $0.4$ kg If a bag produced by this machine is selected at random, find the probability that its weight is less than $49.5$ kg.  [Use, $\varphi (1.5)=0.06681$]