Respuesta :
The events A and B are independent if the probability that event A occurs does not affect the probability that event B occurs.
A and B are independent if the equation P(A∩B) = P(A) P(B) holds true.
P(A∩B) is the probability that both event A and B occur.
Conditional probability is the probability of an event given that some other event first occurs.
P(B|A)=P(A∩B)/P(A)
In the case where events A and B are independent the conditional probability of event B given event A is simply the probability of event B, that is P(B).
Statement 1:A and B are independent events because P(A∣B) = P(A) = 0.12. This is true.
Statement 2:A and B are independent events because P(A∣B) = P(A) = 0.25.
This is true.
Statement 3:A and B are not independent events because P(A∣B) = 0.12 and P(A) = 0.25.
This is true.
Statement 4:A and B are not independent events because P(A∣B) = 0.375 and P(A) = 0.25
This is true.
A and B are independent if the equation P(A∩B) = P(A) P(B) holds true.
P(A∩B) is the probability that both event A and B occur.
Conditional probability is the probability of an event given that some other event first occurs.
P(B|A)=P(A∩B)/P(A)
In the case where events A and B are independent the conditional probability of event B given event A is simply the probability of event B, that is P(B).
Statement 1:A and B are independent events because P(A∣B) = P(A) = 0.12. This is true.
Statement 2:A and B are independent events because P(A∣B) = P(A) = 0.25.
This is true.
Statement 3:A and B are not independent events because P(A∣B) = 0.12 and P(A) = 0.25.
This is true.
Statement 4:A and B are not independent events because P(A∣B) = 0.375 and P(A) = 0.25
This is true.
You can use the definition of dependent and independent events to find out which pair of events are independent and which are not.
Both the events A and B are independent events.
What is the chain rule in probability for two events?
For two events A and B:
The chain rule states that the probability that A and B both occur is given by:
[tex]P(A \cap B) = P(A)P(B|A) = P(B)P( A|B)[/tex]
Which pair of events are called independent events?
When one event's occurrence or non-occurrence doesn't affect occurrence or non-occurrence of other event, then such events are called independent events.
Symbolically, we have:
Two events A and B are said to be independent iff we have:
[tex]P(A \cap B) = P(A)P(B)[/tex]
Thus, from the chain rule of probability, we get
[tex]P(A|B) = P(A)\\P(B|A) = P(B)[/tex]
(assuming A and B are independent events)
Using the above conclusion to find whether A and B are independent or not
Since it is given that we have P(A) = 0.8, P(B) = 0.4, [tex]P(A \cap B) = 0.32[/tex]
Since we have
[tex]\rm P(A \cap B) = 0.32\\and \\P(A)P(B) = 0.8 \times 0.4 = 0.32\\Thus,\\P(A \cap B) = P(A)P(B) = 0.8 \times 0.4[/tex]
thus, A and B are independent events.
Thus, we have:
[tex]P(A|B) = P(A) = 0.8\\P(B|A) = P(B) = 0.4[/tex]
Thus,
Both A and B are independent events.
Learn more here about dependent and independent events here:
https://brainly.com/question/3898488
