Explanation
1. Identify the Given Information
Let the probability of event A occurring be P(A)=x and the probability of event B occurring be P(B)=y.
Since A and B are independent events, the probability of their complements (not occurring) are:
We are given two main conditions:
Probability that both occur:
P(A∩B)=P(A)⋅P(B)=81
x⋅y=81— (Equation 1)
Probability that neither occurs:
P(A′∩B′)=P(A′)⋅P(B′)=83
(1−x)(1−y)=83— (Equation 2)
We are also given the condition: x > y.
2. Simplify the Equations
Expand Equation 2:
1−x−y+xy=83
Substitute the value of xy=81 from Equation 1 into this equation:
1−(x+y)+81=83
Rearrange the terms to find (x+y):
1+81−83=x+y
89−83=x+y
x+y=86=43— (Equation 3)
3. Solve for x and y
We now have a system of two equations:
Sum: x+y=43
Product: xy=81
We can use the algebraic identity (x−y)2=(x+y)2−4xy to find (x−y):
(x−y)2=(43)2−4(81)
(x−y)2=169−84
(x−y)2=169−168=161
Taking the square root on both sides:
x−y=±41
Since we are given that P(A) > P(B) (which means x > y), the difference must be positive:
x−y=41— (Equation 4)
4. Final Calculation
Now, add Equation 3 and Equation 4:
(x+y)+(x−y)=43+41
2x=1⟹x=21
Substitute x=21 back into Equation 3 to find y:
21+y=43
y=43−21=41
Therefore:
Correct Option:
(b) 1/4
