Answer:
neither A nor B will occur simultaneously, as they are mutually exclusive.
Explanation:
To compute the probability that either event A occurs, or B occurs, or both occur, you can use the principle of the union of events. The probability of the union of two events A and B (denoted as A ∪ B) can be calculated as:
�
(
�
∪
�
)
=
�
(
�
)
+
�
(
�
)
−
�
(
�
∩
�
)
P(A∪B)=P(A)+P(B)−P(A∩B)
In this case, you're given:
�
(
�
∁
)
=
0.40
P(A
∁
)=0.40, which means the probability of the complement of A (i.e., the probability that A does not occur).
�
(
�
∣
�
)
=
0.80
P(B∣A)=0.80, which is the conditional probability of B occurring given that A has occurred.
Let's break it down:
�
(
�
∁
)
P(A
∁
) is the probability that event A does not occur, which is
1
−
�
(
�
)
1−P(A).
�
(
�
∣
�
)
P(B∣A) is the conditional probability that event B occurs given that A has occurred.
So, you can calculate
�
(
�
)
P(A) and
�
(
�
)
P(B) as follows:
�
(
�
)
=
1
−
�
(
�
∁
)
=
1
−
0.40
=
0.60
P(A)=1−P(A
∁
)=1−0.40=0.60
Now, you can use the formula for the union of events to calculate
�
(
�
∪
�
)
P(A∪B):
�
(
�
∪
�
)
=
�
(
�
)
+
�
(
�
)
−
�
(
�
∩
�
)
P(A∪B)=P(A)+P(B)−P(A∩B)
But before we calculate
�
(
�
∩
�
)
P(A∩B), note that events A and B are independent, so
�
(
�
∩
�
)
=
�
(
�
)
⋅
�
(
�
∣
�
)
P(A∩B)=P(A)⋅P(B∣A).
�
(
�
∩
�
)
=
�
(
�
)
⋅
�
(
�
∣
�
)
=
0.60
⋅
0.80
=
0.48
P(A∩B)=P(A)⋅P(B∣A)=0.60⋅0.80=0.48
Now, plug this value into the formula:
�
(
�
∪
�
)
=
0.60
+
�
(
�
)
−
0.48
P(A∪B)=0.60+P(B)−0.48
Solve for
�
(
�
)
P(B):
�
(
�
)
=
�
(
�
∪
�
)
+
0.48
−
0.60
P(B)=P(A∪B)+0.48−0.60
�
(
�
)
=
�
(
�
∪
�
)
−
0.12
P(B)=P(A∪B)−0.12
Now, you have the equation:
�
(
�
∪
�
)
=
0.60
+
�
(
�
∪
�
)
−
0.12
−
0.48
P(A∪B)=0.60+P(A∪B)−0.12−0.48
Simplify:
�
(
�
∪
�
)
=
0.60
−
0.12
−
0.48
P(A∪B)=0.60−0.12−0.48
�
(
�
∪
�
)
=
0.00
P(A∪B)=0.00
So, the probability that either event A occurs, or B occurs, or both occur is 0.00. This means that neither A nor B will occur simultaneously, as they are mutually exclusive.