There is a $50\%$ chance that a six-year-old child drops an ice cream that they are eating. Ice creams are given to $5$ six-year-old children and $45$ six-year-old children are divided into nine groups of five and each child is given an ice cream. Calculate the probability that exactly one of the children in at most one of the groups drops their ice cream. [Write your answer correcting to three decimal places]

A coin is biased such that heads is three times as likely as tails on each toss. The coin is tossed $12$ times. The variables $H$ and $T$ are, respectively, the number of heads and the number of tails obtained. Find the value of $\frac{P(H=7)}{P(T=7)}$.

Given that $Q~B(n,0.3)$ and it is given that $P(Q=0)>0.1$, find the greatest possible value of $n$. [Use, $\ln (0.1)=-2.30258509,\ln (0.7)=-0.356674944$]

The variable $T~B(n,0.96)$ and it is given that $P(T=n)>0.5$, find the greatest possible value of $n$. [Use, $\ln (0.1)=-0.693147181,\ln (0.7)=-0.0408219945$]

Given that $R~B(n,0.8)$ and it is given that $P(R>n-1)<0.006$, find the least possible value of $n$. [Use,$\ln (0.006)=-5.11599581,\ln (0.8)=-0.223143551$]

The number of damaged eggs, $D$, in cartons of six eggs have been recorded by an inspector at a packing depot. The following table shows the frequency distribution of some of the numbers of damaged eggs in $150000$ boxes.

The distribution of $D$ is to be modelled by $D~B(6,p)$. Estimate the suitable value of $p$, correcting to four decimal places.

The number of damaged eggs, $D$, in cartons of six eggs have been recorded by an inspector at a packing depot. The following table shows the frequency distribution of some of the numbers of damaged eggs in $150000$ boxes.

The distribution of $D$ is to be modelled by $D~B(6,p)$.Calculate estimates for the value of $a$ and of $b$.

The number of damaged eggs, $D$, in cartons of six eggs have been recorded by an inspector at a packing depot. The following table shows the frequency distribution of some of the numbers of damaged eggs in $150000$ boxes.

The distribution of $D$ is to be modelled by $D~B(6,p)$. Calculate an estimate for the least number of additional cartons that would need to be inspected for there to be at least $8400$ cartons containing one damaged egg.

The number of months during the $4-$month monsoon season (June to September) in which the total rainfall was greater than $5$ metres, $R$, has been recorded at a location in Meghalaya for the past $32$ years, and is shown in the following table.

The distribution of $R$ is to be modelled by $R~B(4,p)$. Find the value of $p$ and state clearly what this value represents. Give a reason why, in real life, it is unlikely that a binomial distribution could be used to model these data accurately.