But change the event A as “the first dice roll is 3”; event B as “the second dic

Question

But change the event A as “the first dice roll is 3”; event B as “the second dice roll is 4” ; event C as “the sum of the dice rolls are 7.”

c) A and D C. 1-8.3 A pair of dice are rolled. Let A be the event of obtaining an odd number on the first die and B be the event of obtaining and odd number on the second die. Let C be the event of obtaining an odd total from both dice. a) Show that A and Bare independent, that A and C are independent, and that B and C are independent. b) Show that A, B, and C are not mutually independent.

Explanation / Answer

probabilty of event 3 on die=P(A)=1/6

probabilty of 4 on second die P(B)=1/6

probabilty of 7 as sum on both dice =P(C) =1/6 {(1,6).(2.5),(3,4),(4,3),(5,2),(6,1)

a)as there is only one event when 3 and 4 appear on first and second die out of 36, hence P(AnB)=1/36

hence P(A|B) =P(AnB)/P(B) =(1/36)/(1/6)=1/6 =P(A) ; hence A and B are independent

there is only one event when 3 appear on first die out of 36, and sum is 36 hence P(AnC)=1/36

similarly P(A||C)=P(AnC)/P(C) =(1/36)/(1/6) =1/6=P(A)

same goes for B, hence B and C are independent,

b) P(AnBnC)=1/36 which is not equal to P(A)*P(B)*P(C) hence not mutually independent

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