Part (a)
\nIf a random variable $X$ follows a binomial distribution with parameters $n$ and $p$. The probability of $r$ successes out of $n$ trials is given by:
\n$P(X=r)=P(r,n)=C^n_{r}p^{r}q^{n-r}$
\nwhere $p$ is the probability of success for each trial and $q$ is the probability of failure for each trial.
\nThe probability that a student cycles to college is $\\var{p}$, therefore $p=\\var{p}$ and $q=1-\\var{p}=\\var{q}$.
\nWe are interested in claculating the probability that none of the sample of $\\var{n}$ students cycle to college so $r=0$ and $n=\\var{n}$
\n$P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}$
\n$P(\\var{r0}, \\var{n})= \\var{pr0}$
\n\n
Part (b)
\nWe are interested in claculating the probability that at least $\\var{r}$ of the $\\var{n}$ students cycle to college. Let $X$ represent the number of students that cycle to college. We need to calculate:
\n$P(X \\geq \\var{r}) = P(X= \\var{r}) + P(X= \\var{r+1})+...+ P(X=\\var{n})$
\n\n
Since $P(X=\\var{r0})+P(X=\\var{r0+1})+...+P(X=\\var{n})=\\var{r0+1}$
\nWe may write
\n$P(X \\geq \\var{r}) = 1-P(X= \\var{r0}) - P(X=\\var{r0+1})-...- P(X=\\var{r-1})$
\n\n
where
\n$P(X= \\var{r0})=P(\\var{r0}, \\var{n})= C^\\var{n}_{\\var{r0}}$ $\\var{p}^\\var{r0}$ $\\var{q}^{\\var{n}-\\var{r0}}=\\var{pr0}$
\n$P(X=1) =P(1, \\var{n})= C^\\var{n}_{1}$ $\\var{p}^{1}$ $\\var{q}^{\\var{n}-1}$ $=\\var{pr1}$
\n$P(X=2) = P(2, \\var{n})=$ $C^\\var{n}_{2}$ $\\var{p}^{2}$ $\\var{q}^{\\var{n}-2}$ $=\\var{pr2}$
\n\nThen
\n$P(X \\geq \\var{r}) = 1-\\var{qn}-\\var{pr1}-\\var{pr2}=\\var{answer2}$
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\nIt is estimated that $\\var{p_perc}$% of all CIT students cycle to college. A random sample of $\\var{n}$ CIT students is chosen.
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