Conditional probabilities in a Bayesian Network

Question

The Bayesian Network LMV has three nodes for boolean variables, L, M and V.

Bayesian Network LMV$$\require{enclose}\enclose{circle}{L}\lower{2ex}{\searrow\lower{2ex}{\enclose{circle}{V}}\swarrow}\enclose{circle}{M}$$

The probabilities for L and M are:

P(M = true) = 0.2

P(L = true) = 0.7

The conditional probabilities for variable V:

conditional probabilities for V $$\begin{array}{l|l|l}L&M&\mathsf P(V\,{=}\,\text{true}\mid L, M)\\\hline \text{true}&\text{true}&0.9\\\text{true}&\text{false}&0.5\\\text{false}&\text{true}&0.3\\\text{false}&\text{false}&0.05\end{array}$$

What is the value of P(V = false | L = false)?

I am trying to use Bayes Theorem to get the result, but i just can't seem to get it right.

Answer 1

Probabilities:

P(M) = 0.2

P(¬M) = 0.8

P(L) = 0.7

P(¬L) = 0.3

Solution:

P(¬V│¬L) = P(¬V,¬L)/P(¬L)

P(¬V│¬L) = P(¬V,¬L)/0.3

We use marginal probability to find the probability of V = false, when L = false and M = true or false:

¬V|¬L,M = 1-0.3 = 0.7

¬V|¬L,¬M = 1-0.05 = 0.95

P(¬V│¬L) = ∑ P(¬V,¬L, M=m)/0.3

P(¬V│¬L) = (P(¬V|¬L, M=t) + P(¬V|¬L, M=f))/0.3

P(¬V│¬L) = ((¬V * ¬L * M) + (¬V * ¬L * ¬M))/0.3

P(¬V│¬L) = ((0.7 * 0.3 * 0.2) + (0.95 * 0.3 * 0.8))/0.3

P(¬V│¬L)=0.9