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.

In a particular country, $90\%$ of both females and males drink tea. Of those who drink tea, $40\%$ of the females and $60\%$ of the males drink it with sugar. Find the probability that in a random selection of two females and two males, all four people drink tea.

In a particular country, $90\%$ of both females and males drink tea. Of those who drink tea, $40\%$ of the females and $60\%$ of the males drink it with sugar. Find the probability that in a random selection of two females and two males, an equal number of females and males drink tea with sugar.  [Write your answer correcting to three decimal places]

It is estimated that $0.5\%$ of all left-handed people and $0.4\%$ of all right-handed people suffer from some form of colour-blindness. A random sample of $200$ left-handed and $300$ right-handed people is taken. Find the probability that there is exactly one person in the sample that suffers from colour-blindness.[Write your answer correcting to three decimal places]