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I am trying to understand what does Marginal probability mean in Probability theory. I looked it up in wikipedia but its language and notations are way beyond comprehension for somebody without a math background. So far from what I've read I assume that Marginal probability is an unconditional probability, it does not depend on the probability of some other event (ok, but an example could save the day).

I do however have another definition. "Marginal probability -- is a sum of probabilities of mutually exclusive events P(A) = ∑ p(A,B)." I don't understand the formula above however. p(A, B) is an intersection and if A and B are mutually exclusive then p(A, B) = 0. So I am confused completely now.

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If $X$ and $Y$ are random variables taking on values $x_1, x_2, \ldots, x_m,$ and $y_1, y_2, \ldots, y_n,$ respectively, then the joint probabilities $P\{X = x_i, Y = y_j\}$ (meaning the probability that $X$ has value $x_i$ and simultaneously $Y$ has value $y_j$) can be displayed as an array (a matrix, if you like) with $m$ columns and $n$ rows:

$$\scriptstyle\begin{array}{c|c|c|c|c|c} & x_1 & \cdots & x_i & \cdots & x_m&\\\hline y_1& P\{X = x_1, Y = y_1\}& \cdots&P\{X = x_i, Y = y_1\}&\cdots &P\{X = x_m, Y = y_1\}&P\{Y=y_1\}\\\hline \vdots & \vdots&\ddots&\vdots&\ddots&\vdots&\vdots\\\hline y_j&P\{X = x_1, Y = y_j\}&\ddots&P\{X = x_i, Y = y_j\} &\ddots&P\{X = x_m, Y = y_j\}&P\{Y=y_j\}\\\hline \vdots&\vdots&\ddots&\vdots&\ddots&\vdots&\vdots\\\hline y_n&P\{X = x_1, Y = y_m\}&\cdots&P\{X = x_i, Y = y_m\}&\cdots &P\{X = x_m, Y = y_n\}&P\{Y=y_n\}\\\hline &P\{X=x-1\}&\cdots&P\{X = x_i\}&\cdots&P\{X = x_m\}& 1 \end{array}$$ Notice that in addition to the labels $x_1, x_2, \ldots, x_m,$ and $y_1, y_2, \ldots, y_n,$ along the top and on the left respectively, I have added a rightmost ($(m+1)$-th) column and a bottommost ($(n+1)$-th) row. The entries here are the row sums and column sums respectively: the sum of the entries in the $i$-th column is $$P\{X=x_i, y=y_1\}+\cdots+P\{X=x_i, Y=y_j\}+\cdots+P\{X=x_i, Y=y_n\} = \sum_{j=1}^n P\{X=x_i,Y=y_j\}$$ which is just the marginal probability $P\{X = x_i\}$. (The reason for calling this a marginal probability is that it is written down in the margins of the array: here, marginal is not used in the other meaning of the word as something that barely meets the standards or barely gets a passing grade).

Note that the formula $\displaystyle P\{X = x_i\} = \sum_{j=1}^n P\{X=x_i,Y=y_j\}$ is just your P(A) = ∑ p(A,B) where, as noted by DeepakML, the sum is over events $B$ that are mutually exclusive: $Y$ cannot simultaneously have two different values $Y_i$ and $Y_k$, and exhaustive: $Y$ must take on one of the values $y_1, y_2, \ldots, y_n$. In other words, the mutual exclusivity does not apply to $A$ and $B$ as you appear to think.

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  • $\begingroup$ Nice, thorough answer, @Dilip Sarwate. $\endgroup$ – StatsStudent Feb 5 '15 at 16:29
  • $\begingroup$ This is fine for discrete distributions but doesn't explain the continuous cases. $\endgroup$ – Michael R. Chernick Apr 23 '17 at 17:44
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The summation in this case is on B, which is a set of mutually exclusive events and exhaustive.

Thanks

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