use the normal distribution to approximate

a discrete probability distribution. As it happens, the normal distribution becomes an

increasingly better approximation to the binomial distribution as n $->\backslash$infty $.$

$($a$)$ Consider a binomial random variable X where n $=25$ and p $=0\mathrm{.}4.$ What is the

probability that X $=10?$

$($b$)$ Let$$s approximate this random variable X with Y $$ Normal$($np$,$ npq$),$ where q $=$

$1$ p$.$ Notice that Y has the same mean and variance as X$.$ As an approximation

to the true probability that X $=10($found in part $($a$)),$ what is the probability that

Y in $[9\mathrm{.}5,10\mathrm{.}5]?$

$($c$)$ Why did we select an interval to approximate the precise event where X $=10?$

$($d$)$ What is the relative error$2$ of this probability?