Question

In: Statistics and Probability

Let A, B and C be events. Express for k = 0, 1, 2, 3, the...

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.

Solutions

Expert Solution

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

orchestra answered 1 year ago

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