Question

Assume XX has a binomial distribution. Use the binomial formula, tables, or technology to calculate the probability of the indicated event:

a. n=22, p=0.7n=22, p=0.7

P(14 ≤ X ≤ 17)=P(14 ≤ X ≤ 17)=

Round to four decimal places if necessary

b. n=25, p=0.4n=25, p=0.4

P(9 < X < 12)=P(9 < X < 12)=

Round to four decimal places if necessary

Answer 1

Here we use the technology R- software to calculate the probabilities, The r-codes are provided along with output

a)

> n=22
> p=0.7
> p1=dbinom(14,n,p)+dbinom(15,n,p)+dbinom(16,n,p)+dbinom(17,n,p)
> round(p1,4)
[1] 0.649

Or

> p1=pbinom(17,n,p)-pbinom(13,n,p) # the 13 is not included it will start from 14 but 17 is included
> round(p1,4)
[1] 0.649

b)

> n=25
> p=0.4
> p1=dbinom(10,n,p)+dbinom(11,n,p) # becuase both 9 and 12 are exclusive, so we calculate for 10 and 11 only
> round(p1,4)
[1] 0.3077

or

> p1=pbinom(11,n,p)-pbinom(9,n,p) # here both bound are exclusive, therefore we took 11 and the command is also exclusive for lower bound so we took 9 as it is.
> round(p1,4)
> round(p1,4)
[1] 0.3077

i.e.,

a)

n=22, p=0.7

P(14 ≤ X ≤ 17)= 0.649

b)

n=25, p=0.4

P(9 < X < 12)= 0.3077

( If you need by hand calculation please comment)