In: Statistics and Probability
Let A, B and C be events. Express for k = 0, 1, 2, 3,
the probabilities that:
a) Exactly k of events A, B and C occur.
b) At least k of events A, B and C occur.
An event can occur or not occur so there are 2 cases for each event.
Total cases for 3 events = 2*2*2 = 8
a) k = 0
P( Exactly 0 event occur ) = Selecting 0 event from 3 / Total cases
= 3C0 / 8 = 1/8 = 0.125
k = 1
P( Exactly 1 event occur ) = Selecting 1 event from 3 / Total cases
= 3C1 / 8 = 3/8 = 0.375
k = 2
P( Exactly 2 event occur ) = Selecting 2 events from 3 / Total cases
= 3C2 / 8 = 3/8 = 0.375
k = 3
P( Exactly 3 event occur ) = Selecting 3 events from 3 / Total cases
= 3C3 / 8 = 1/8 = 0.125
b) Atleast k events means exactly k,k+1,K+2 and so on
k= 0
P( Atleast 0 event occur ) = Selecting 0,1,2,3 event from 3 / Total cases
= 3C0 + 3C1+ 3C2 + 3C3 / 8 = 8/8 = 1
k=1
P( Atleast 1 event occur ) = Selecting 1,2,3 event from 3 / Total cases
= 3C1+ 3C2 + 3C3 / 8 = 7/8 = 0.875
k=2
P( Atleast 2 event occur ) = Selecting 2,3 event from 3 / Total cases
= 3C2 + 3C3 / 8 = 4/8 = 0.5
k=3
P( Atleast 3 event occur ) = Selecting 3 event from 3 / Total cases
= 3C3 / 8 = 1/8 = 0.125
If any doubt please feel free to ask through the comments section. Thank You