Question

Which of the following are true if events A and B are independent? Select all that apply.

A. P(A | B) = P(A)
B. P(A | B) = P(B)
C. P(A | B) = P(A and B)
D. P(B | A) = P(A and B)
E. P(B | A) = P(A)
F. P(B | A) = P(B)

Answer 1

In the context of independent events, the correct statements are that P(A | B) = P(A) and P(B | A) = P(B), indicating that the occurrence of one event does not affect the probability of the other event occurring. Other options presented do not accurately represent the properties of independent events in probability.

Step-by-step explanation:

When assessing whether events A and B are independent, it is essential to understand the criteria for independence in probability theory. Specifically, two events are independent if the probability of one event occurring does not affect the probability of the other event occurring. This can be mathematically represented as follows: P(A AND B) = P(A)P(B), P(A|B) = P(A), and P(B|A) = P(B).

If events A and B are independent, the correct statements among the choices provided are:

• 
  
• 
  

Option A is true because if A and B are independent, the probability of A occurring given that B has occurred is the same as the probability of A occurring on its own.

Option F is also correct for the same reason applied to event B; the probability of B occurring given that A has occurred is the same as the probability of B occurring on its own.

The remaining options are incorrect because they do not align with the definition of independent events:

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• 
  
• 
  
• 
  

Answer 2

The correct statement are (A) and (F).

Step-by-step explanation:

Events A and B are independent or mutually independent events if the chance of their concurrent happening is equivalent to the multiplication of their distinct probabilities.

That is,

`P(A\cap B)=P(A)* P(B)`

The conditional probability of event A given B is computed using the formula:

`P(A|B)=(P(A\cap B))/(P(B))`

And the formula for the conditional probability of event B given A is:

`P(B|A)=(P(A\cap B))/(P(A))`

Consider that events A and B are independent.

Then the conditional probability of event A given B will be:

`P(A|B)=(P(A\cap B))/(P(B))`

`=(P(A)* P(B))/(P(B))\\\\=P(A)`

And the conditional probability of event B given A will be:

`P(B|A)=(P(A\cap B))/(P(A))`

`=(P(A)* P(B))/(P(A))\\\\=P(B)`

Thus, the correct statement are (A) and (F).