Conditional Probability Help?

Question

P(A) = 0.25, P(B) = 0.30, P(C) = 0.48, P(D) = 0.35
P(A n B) = 0.05, P(B n c) = 0.10, P(A n c) = 0, P(B n D) = 0

*D is a special case of C

1. The probability that has B but not A?
2. The probability does not have B but has A?
3. Does having D depend on whether or not patient has A?
4. Does having D depend on whether or not patient has C?

Answer 1

I'll use Ac, Bc, Cc, Dc to represent the complements (not A, not B, etc).

1. P(B) = P(B n A) + P(B n Ac)
<=> 0.30 = 0.05 + P(B n Ac)
So P(B but not A) = P(B n Ac) = 0.25.

2. P(A) = P(A n B) + P(A n Bc)
<=> 0.25 = 0.05 + P(A n Bc)
Sp P(A but not B) = P(A n Bc) = 0.20.

3. If D is a special case of C, then P(A n D) <= P(A n C) = 0, so P(A n D) = 0. Since this is not the same as P(A) . P(D) they are not independent, so the answer is Yes.

4. Obviously yes, since D is a subset of C. More formally, since D is a special case of C, P(D | Cc) = 0 which is different to P(D). So D is dependent on C.

Note that all the following statements are, individually, equivalent to the statement that X and Y are independent:
P(X n Y) = P(X) . P(Y)
P(X | Y) = P(X)
P(X | Yc) = P(X)
P(X | Y) = P(X | Yc)
and the same with X and Y swapped, plus any equivalent statement with X and Xc swapped, or Y and Yc swapped, or both; for instance, one of these equivalent statements is P(Yc | X) = P(Yc).

Answer 2

You need to know some important formulas,

For any event A:
P(A) = 1 - P(not A)

For any two events A and B:
P(A ∩ B) = P(A) + P(B) - P(A U B)

(Conditional probability) For any two events A and B, with P(B) > 0:
P(A | B) = P(A ∩ B) / P(B)

(Law of total probability)
P(B) = P(B ∩ A) + P(B ∩ notA)

Two events A and B are independent if and only if:
P(A|B) = P(A)


Since D is a special case of C, then
P(C ∩ D) = P(D) and P(C U D) = P(C)


1) Find P( B ∩ notA ) = P(B) - P(B ∩ A)
P( B ∩ notA ) = 0.30 - 0.05 = 0.25

2) Find P(A ∩ notB) = P(A) - P(A ∩ B)
P(A ∩ notB) = 0.25 - 0.05 = 0.20

3) Does P(D|A) = P(D)?
P(D|A) = P(D ∩ A) / P(A)
Since D is included in C, P(D ∩ A) ≤ P(A ∩ C) = 0
since P(D ∩ A) = 0 the events are defined to be independent as there is no overlap.

4) Does P(D|C) = P(D)?
P(D ∩ C) / P(C) = P(D) / P(C) ≠ P(D) so yes, having D is dependent on having C.

Answer 3

q1
0.3-0.05=0.25

q2
0.25-0.05=0.20

q3
no, because P(A n c) = 0, and D's inside C

q4
yes, D's subset to C