Question

Question 7: The following information is given about events A and B: ??((?? ?? ? ?? ?? ) ?? ) = ??. ?? ??(?? ?? ? ??) = ??. ?? ??(??) = ??. ?? Based on this information, determine whether A and B are independent or dependent with respect to each other

Answer 1

We are given here that:

P ( ( A^c  ? B^c)^c ) = 0.3
P( A^c ? B ) = 0.3
P(A) = 0.6

Let the Venn Diagram for the 2 events be given as:

Note that from the Venn diagram, if we see, we have:

A^c would be the area yellow + grey
B^c would be the area blue + grey
Therefore,  A^c  ? B^c would be the area yellow + blue+ grey

Therefore, ( ( A^c  ? B^c)^c ) would be the green area.

Therefore P( green ) = 0.3

Next, we can see that:

A^c would be the area yellow + grey
B is the area green + yellow
Therefore, A^c ? B would be the yellow area.
P( yellow ) = 0.3

P(A) = 0.6, that is blue + green

P(Blue ) = 0.6 - 0.3 = 0.3

Therefore, P(B) = green + yellow = 0.6

P( A) = 0.6

P(A)P(B) = 0.6*0.6 = 0.36

P( A ? B) = green area = 0.3

Now as P( A ? B) is not equal to P(A)P(B), therefore A and B are not independent events and are dependent events.