Conditional Probability

[sessomw5098]

Given the following:
P(shop) = 0.25
P(rob) = 0.02
P(enter-store | !shop, !rob) = 0.01
P(enter-store | shop, !rob) = 0.80
P(enter-store | !shop, rob) = 0.50
P(enter-store | shop, rob) = 0.90
P(point-gun | rob) = 0.70
P(point-gun | !rob) = 0.01

Find:
a) P(enter-store)
b) P(shop | enter-store)
c) P(rob | enter-store)
d) P(shop | enter-store, point-gun)
e) P(rob | enter-store, point-gun)

The problem I'm having is given P(A|B, C), how to extract P(B) or even P(C), and so on.

Can I say: P(shop)P(rob) = .005, then P(enter-store|.005)? How do I get P(enter-store)?

Thanks,
Jay

 

[Valkarie]

a) P(enter-store) = P(enter-store | shop, !rob)P(shop) + P(enter-store | !shop, rob)P(rob) + P(enter-store | !shop, !rob)P(!shop, !rob) = 0.80 × 0.25 + 0.50 × 0.02 + 0.01 × (1 - 0.25 - 0.02) = 0.2025b) P(shop | enter-store) = P(enter-store | shop, !rob)P(shop) / P(enter-store) = 0.80 × 0.25 / 0.2025 = 0.4950c) P(rob | enter-store) = P(enter-store | !shop, rob)P(rob) / P(enter-store) = 0.50 × 0.02 / 0.2025 = 0.0990d) P(shop | enter-store, point-gun) = P(enter-store | shop, rob)P(shop)P(point-gun | rob) / P(enter-store, point-gun) = 0.90 × 0.25 × 0.70 / (P(enter-store | shop, rob)P(shop)P(point-gun | rob) + P(enter-store | !shop, rob)P(rob)P(point-gun | !rob)) = 0.90 × 0.25 × 0.70 / (0.90 × 0.25 × 0.70 + 0.50 × 0.02 × 0.01) = 0.9755e) P(rob | enter-store, point-gun) = P(enter-store | !shop, rob)P(rob)P(point-gun | !rob) / P(enter-store, point-gun) = 0.50 × 0.02 × 0.01 / (P(enter-store | shop, rob)P(shop)P(point-gun | rob) + P(enter-store | !shop, rob)P(rob)P(point-gun | !rob)) = 0.50 × 0.02 × 0.01 / (0.90 × 0.25 × 0.70 + 0.50 × 0