The events E1 and E2 are not independent events.
How to determine if the events are independent?
The two events are given as:
E1 and E2 such that
P(E1) = 0.27, P(E2) = 0.40 and P(E1 U E2) = 0.58.
The two events E1 and E2 are independent if the following equation is true
P(E1 and E2) = P(E1) * P(E2)
Where
P(E1 and E2) = P(E1) + P(E2) - P(E1 U E2)
Substitute the known values in the above equation
P(E1 and E2) = 0.27 + 0.40 - 0.58
Evaluate the sum in the above equation
P(E1 and E2) = 0.67 - 0.58
Evaluate the difference in the above equation
P(E1 and E2) = 0.09
Substitute P(E1 and E2) = 0.09 in P(E1 and E2) = P(E1) * P(E2)
P(E1) * P(E2) = 0.09
Substitute P(E1) = 0.27 and P(E2) = 0.40 in the above equation
0.27 * 0.40 = 0.09
Evaluate the product
0.108 = 0.09
The above is false because 0.108 and 0.09 are not equal
Hence, the events E1 and E2 are not independent events.
Independent Events A and B
How to solve for x
We have:
P(A) = x
P(B) = x + 0.2
P(A n B) = 0.15
For independent events, we have:
P(A) * P(B) = P(A n B)
So, we have:
x * (x + 0.2) = 0.15
Expand
x^2 + 0.2x - 0.15 = 0
Using a graphing calculator, we have:
x = 0.3
Hence, the value of x is 0.3
The value of P(A U B)
This is calculated using
P(A U B) = P(A) + P(B) - P(A n B)
Substitute the known values in the above equation
P(A U B) = x + x + 0.2 - 0.15
Evaluate the like terms
P(A U B) = 2x + 0.05
Substitute 0.3 for x
P(A U B) = 2 * 0.3 + 0.05
Evaluate
P(A U B) = 0.65
Hence, the value of P(A U B) is 0.65
The value of P(A' U B')
This is calculated using
P(A' U B') = P(A') + P(B') - P(A' n B')
Where
P(A') = 1 - x
P(B') = 1 - x - 0.2 = 0.8 - x
P(A' n B') = P(A') * P(B')
P(A' n B') = (1 - x) * (0.8 - x)
Substitute the known values in the equation P(A' U B') = P(A') + P(B') - P(A' n B')
P(A' U B') = (1 - x) + (0.8 - x) - (1 - x) * (0.8 - x)
Substitute 0.3 for x
P(A' U B') = (1 - 0.3) + (0.8 - 0.3) - (1 - 0.3) * (0.8 - 0.3)
Evaluate
P(A' U B') = 0.85
Hence, the value of P(A' U B') is 0.85
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