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The proportion of times an event occurs out of a large number of trials is the likelihood of the event.

The binomial dispersion for a discrete arbitrary variable with two potential results.

A random variable is said to have a binomial dispersion assuming the likelihood mass capacity is:

$P(X=x)=\frac{n!}{(n鈭�x)!x!}{p}^x(1鈭�p{)}^{n鈭�x}$

$X~B(n,p)$

the probability distribution for \(X\) is,

$X~B(80,0.613)$

i. Calculating the mean

we know, the binomial distribution is $X~B(n,p)$

Mean \(=np\)

$=80脳0.613=49.04$

ii. The standard deviation is,

$SD=\sqrt{np(1鈭�p)}$

$鈬�\sqrt{80脳0.613脳(1鈭�0.613)}=4.3564$

Using calculator,

First, we press $2nd$and then press VARS and Scroll down to \(binompdf(n,p,x)\)

n = no. of trials

p = success probability

x = required success

$\text{binompdf}(80,0.613,60)=0.0036$

$P(x=60)=0.0036$

As we know,$P(x>50)=1鈭�P(x鈮�50)$

First, we press $2nd$and then press VARS and Scroll down to \(binompdf(n,p,x)\)

Using,

$1鈭�\text{binomcdf}(80,0.613,50)=0.3718$

Hence,$P(x>50)=0.3718$