Question

In: Advanced Math

Discrete Mathematics: Choose the correct choices. There could be more than one answer: Events A and...

Discrete Mathematics: Choose the correct choices. There could be more than one answer:

Events A and B are independent events if(choose all correct answers). Note: P(A) denotes probability of event A.

a) P(A intersection symbol B)=P(A|B)P(B)

b) P( A intersection symbol B)=P(A)P(B)

c) P( A intersection symbol B)=P( B intersection symbol A)

d) P(A|B)=P(B|A)

Solutions

Expert Solution

Solution:
Let A and B be two events connected to a given random
experiment. If P(B) not equal to 0 then P(A|B) can be defined and in this
case if P(A | B)= P(A), then we can say. that the probability of A
does not depend on the happening of B, i.e., there is one kind of
independence between A and B. Also if P(A) not equal to 0, then P(B|A)
can be defined and in this case if P(B| A)=P(B), we can say that
the probability of B does not depend on the happening of A, i.e;
there is one kind of independence between A and B.We observe that
P(AIB)=P(A), P(BI A)=P(B).
both lead to P(A intersection B)= P(A) P(B) [as from definition P(A|B)=P(A intersection B)/P(B)]
So formally we can define independence of two events as
follows:
Two events A, B are said to be stochastically independent
or statistically independent or simply independent if and only if
P(AB)=P(A)P(B).
Therefore option (a) is correct as P(A|B)=P(A)
and option (b) is also correct (both discussed above)


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