A company finds that one out of four employees will be late

A company finds that one out of four employees will be late to work on a given day. If this company has 41 employees, find the probabilities that the following number of people will get to work on time. (Round your answers to 4 decimal places.

(a) Exactly 30 workers or exactly 34 workers.

(b) At least 26 workers but fewer than 33 workers.

(c) More than 24 workers but at most 35 workers.

Solution

This is a binomial probability :

x is bin(41,0.75)

P(x) = 41Cx *0.75^x *0.25^(41-x)

a)

Exactly 30 workers or exactly 34 workers.
P(x = 30) = 41C30 *0.75^30 *0.25^(41-30) =0.1345
P(x = 34) = 41C34 *0.75^34 *0.25^(41-34) =0.0775
Therefore,

P(X=30 OR X=34) = 0.1345+0.0775 = 0.212 Answer

(b)

At least 26 workers but fewer than 33 workers.
P(26 x < 33) = P(X=26)+P(X=27)+.............+P(X=32)

=P(X<33) - P(X<26)

= 0.7296 - 0.0333

= 0.6963 Answer

(c)

More than 24 workers but at most 35 workers.
P(24 < x 35) = P(X=25)+P(X=26)+.............+P(X=35)

=P(X<=35) - P(X<=24)

= 0.9635 - 0.0152

= 0.9483 Answer