Q. 14.8( 5 Votes )

For a loaded die, the probabilities of outcomes are given as under:

P (1) = P (2) = 0.2, P (3) = P (5) = P (6) = 0.1 and P (4) = 0.3.

The die is thrown two times. Let A and B be the events, same number each time and a total score is 10 or more respectively. Determine whether or not A and B are independent.

Answer :

Given that for a loaded die-


P (1) = P (2) = 0.2, P (3) = P (5) = P (6) = 0.1 and P (4) = 0.3


And given that die is thrown two times and A is the event of same number each time


B is the event of a total score is 10 or more.


So,


A = {(1,1), (2,2), (3,3), (4,4), (5,5), (6,6)}


P(A)= [P (1,1) +P (2,2) +P (3,3) + P (4,4) + P (5,5) + P (6,6)]


P(A)= [P (1) × P (1) + P (2) × P (2) + P (3) × P (3) + P (4) × P (4) +P (5) × P (5) + P (6) × P (6)]


P(A)= [0.2×0.2+ 0.2×0.2+ 0.1×0.1+0.3×0.3+ 0.1×0.1+ 0.1


×0.1]


P(A)= [0.04+ 0.04+ 0.01+ 0.09+ 0.01+0.01]


P(A)= [0.20]


&


B= EVENT OF TOTAL SCORE IS 10 OR MORE


B= {(4,6), (5,5), (5,6), (6,4), (6,5), (6,6)}


P(B)= [P (4,6) + P (5,5) + P (5,6) + P (6,4) + P (6,5) + P (6,6)]


P(B)= [P (4) ×P (6) + P (5) ×P (5) + P (5) ×P (6) + P (6) ×P (4) + P (6) ×P (5) + P (6) ×P (6)]


P(B)= [0.3×0.1+ 0.1×0.1+ 0.1×0.1+ 0.1×0.3+ 0.1×0.1+ 0.1×0.1]


P(B)= [0.03+ 0.01+ 0.01+ 0.03+ 0.01+ 0.01]


P(B)= [0.10]


ALSO, probability of an intersection B (i.e. both the events occur simultaneously)


A Ո B = {(5,5), (6,6)}


HENCE,


P (A Ո B) = P (5,5) + P (6,6)


P (A Ո B) = P (5) × P (5) + P (6) ×P (6)


P (A Ո B) = 0.1×0.1+ 0.1×0.1


P (A Ո B) = 0.01+0.01


P (A Ո B) = 0.02


We know that if two events are independent than


P (A Ո B) = P(A) P(B)


HERE,


P(A). P(B)= 0.20× 0.10 = 0.02


SO, P (A Ո B) = P(A) P(B)


Hence, A and B are independent events.


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