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I'm currently reading up to try and complete an assignment but our lecture notes are very sparse and skip over the basics to instantly move onto solving random questions. Because of this I'm having trouble piecing together my understanding of things. This is a really simple question but how do I read a probability distribution expressed as a matrix?

If I was given the following:

$f_A$ = $\begin{bmatrix}0.2 & 0.8\ \end{bmatrix}$
$f_{B|A}$ = $\begin{bmatrix}0.4 & 0.6\\ 0.3 & 0.7 \end{bmatrix}$

What do these matrices mean? Is the first number in a row the probability of success and the second the failure or vice versa? What about when there are two rows?

I just need to make sure I'm understanding things correctly.

Constantconfusion
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    The first is the marginal probabilities of $A$ (if the values are $A=1$ vs. $A=2$, this would be $(\Pr[A=1],\Pr[A=2])$. The matrix contains in entry $(i,j)$ the conditional probability of $A=j$ given $B=i$. You know that is the case because the some of the rows is 1. – Learner Apr 09 '13 at 15:20
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    Depending on what $A$ and $B$ mean, you are either being told that they are random variables, each taking on two values, say $1$ and $2$, or they are events with $P(A)=0.2$, $P(A^c)=0.8$ or vice versa depending on the convention used in your class. In the latter case, the rows of the matrix $f_{B\mid Z}$ then correspond to conditional probabilities $P(B\mid A)$, $P(B^c\mid A)$ and $P(B\mid A^c), P(B^c \mid A^c)$ respectively. – Dilip Sarwate Apr 09 '13 at 15:24
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    If you are beginning to study Markov chains, the matrix is a transition matrix. At any time, the system is in one of two states, which we can call State $1$ and State $2$. If we are now in State $1$, the probability that the "next" time we will be in State $1$ is $0.4$, and the probability we will be in State $2$ is $0.6$. If we are currently in State $2$, the probability that next time we will be in State $1$ is $0.3$, and the probability we will be in State $2$ is $0.7$. – André Nicolas Apr 09 '13 at 15:26
  • Thanks I think I get it now. But what do I do when set values aren't provided? The question that these distributions are from just state that "Probability distributions have been determined to give the probability that there is a commercial quantity of oil available". Do I simply assume that the entry at i = 0 or j = 0 is the probability that there is a commercial quantity of oil available? – Constantconfusion Apr 09 '13 at 15:26
  • @Constantconfusion Basically, these are just binary (nominal) random variables. What value you assign to them is mostly immaterial. You could call it (0 vs. 1), (1 vs. 2), (red vs. blue), (good vs. bad) etc... There are though some cases where this could matter... – Learner Apr 09 '13 at 15:31

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