1. Define the Probability Relation:
Since the events A,B,C, and D are mutually exclusive and exhaustive, the sum of their probabilities is equal to 1.
P(A)+P(B)+P(C)+P(D)=1
Let the given ratio be equal to a constant k:
2P(A)=3P(B)=5P(C)=8P(D)=k
2. Solve for k:
We can express the probabilities as:
P(A)=2k,P(B)=3k,P(C)=5k,P(D)=8k
Substituting these into the summation equation:
2k+3k+5k+8k=1
18k=1⟹k=181
3. Calculate Individual Probabilities:
P(A)=182
P(B)=183
P(C)=185
P(D)=188
4. Find Geometric Mean (G):
The geometric mean of four numbers is the fourth root of their product:
G=(P(A)⋅P(B)⋅P(C)⋅P(D))41
G=(182⋅183⋅185⋅188)41
G=(1842⋅3⋅5⋅8)41=18(240)41
5. Calculate 9G:
9G=9⋅18(240)41=2(240)41
Since 240=16⋅15, and 1641=2:
9G=2(16⋅15)41=22⋅(15)41=1541
Correct Option:
(b) 1541
