We choose a month of the year in such a manner that each mon

We choose a month of the year, in such a manner that each month has the same probability. Find out weather the following events are independant: a) the events \"outcome is an even numbered month\" (i.e., Feb, April, June, etc.) and \"outcome is first half of the year\". b) teh events \"outcome is an even numbered month\" (i.e., February, April, June, etc.) and \"outcome is a summer month\" (i.e., June, July, August).

Solution

(a) for indepenedent events,

P(A and B)=P(A).P(B)

P(A)=Probability of having even numbered month i.e. Feb, Apr, June, Aug, Oct, Dec=6/12=1/2

P(B)=Probability outcome is first half of the year i.e. Jan,Feb,March,Apr,May,June=6/12=1/2

P(A and B)=Probability of having an even numbered and month in first half of the year i.e. Feb,Apr,June=3/12=1/4

P(A).P(B)=(1/2)*(1/2)=1/4

Since P(A and B) is equal to P(A).P(B), hence these are independent events

(b)

P(A)=Probability of having even numbered month i.e. Feb, Apr, June, Aug, Oct, Dec=6/12=1/2

P(B)=Probability outcome is a summer month i.e. June, July,Aug=3/12=1/4

P(A and B)=Probability of having an even numbered and summer month i.e. June, Aug=2/12=1/6

P(A).P(B)=(1/2)*(1/4)=1/8

Since P(A and B) is not equal to P(A).P(B), hence these are not independent events