Explanation
To find the value of P(C), we follow these steps:
1. Define the Variables
Let 4P(A)=2P(B)=P(C)=k.
This gives us the following probabilities:
2. Apply Conditions
Disjoint Events: Since A and B are disjoint, P(A∩B)=0. Consequently, the intersection of all three events P(A∩B∩C)=0.
Independent Events: A and C are independent, so P(A∩C)=P(A)P(C)=(4k)(k)=4k2.
Independent Events: B and C are independent, so P(B∩C)=P(B)P(C)=(2k)(k)=2k2.
3. Use the Union Formula
The formula for the union of three events is:
P(A∪B∪C)=P(A)+P(B)+P(C)−P(A∩B)−P(A∩C)−P(B∩C)+P(A∩B∩C)
Substituting the known values and conditions:
5P(A)=P(A)+P(B)+P(C)−0−P(A)P(C)−P(B)P(C)+0
5(4k)=4k+2k+k−4k2−2k2
4. Solve for k
Multiply the entire equation by 4 to simplify the denominators:
5k=k+2k+4k−k2−2k2
5k=7k−3k2
3k2=2k
Since k (the probability) cannot be 0 for a valid event C, we can divide by k:
3k=2
k=32
Since P(C)=k, then P(C)=32.
Correct Option:
(d) 32
