Question

Suppose p(a) = 0.76, p(b/a)= 0.30 and p(b/a bar)= 0.02. Find p(a and b), and p(a bar and b). use these to construct a probability table. Now use the table to find the following.

a) p(b bar/ a)
b) p( b bar/ a bar)
c) p(b)
d) p( b bar)
e) p( a / b)
f) p( a bar/ b)
g) p( a/ b bar)
h) p( a bar/ b bar)

Answer 1

Properties you can use
\[
(1)\: P(\bar B | A) = 1 - P(B | A) \\
(2)\: P(A|B) = P(B|A)P(A)/P(B)
(3)\: P(A|B) = P(A\cap B)/P(B)
\]
Not true though:
\[ P(A|\bar B) = 1 - P(A|B)\].
This one would require you first to switch both events using (2):
\[P(A|\bar B) = P(\bar B|A)P(A)/P(\bar B)\]
and \(P(\bar B) = 1- P(B)\).

Answer 2

again, with better formatting.
\[(1)\: P(\bar B | A) = 1 - P(B | A) \\
(2)\: P(A|B) = P(B|A)P(A)/P(B)\\
(3)\: P(A|B) = P(A\cap B)/P(B)\]

Answer 3

another one
\[(4)\: P(A) = P(A \cap B) + P(A\cap \bar B)\]

Answer 4

with these, you can solve most or all of them

Answer 5

i am still not sure how to solve. can you help me?