How do you calculate marginal, joint, and conditional probabilities from a two-way table?

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Answer 1 · 2015-05-18T20:29:45

If you are given a pmf = $p_(XY)(x,y)$

and you would like to find the marginal $p_Y(y)$

we would use the formula $p_y(y) = sum_ip(x_i,y)$

in other words you would sum over all of $x$ at the point $y$

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So if we look at this table and want to find the marginal $p_Y(3)$

we go:

$p_Y(3) = P( Y = 3)$
$= P(Y = 3, X = 3) + P(Y = 3, X=4)$
$= 0.1 + 0.2$
$=0.3$

Now to look at the formula for the conditional probability

we can look at the formula for $x$ given $y$ which is a conditional probability.

$p_(X|Y)(x|y) = P(X = x_i | Y = y_j) = (P(X = x_i, Y= y_j))/(P(Y = y_j))$

$=(p_(XY)(x_i,y_j))/(p_Y(y_i))$

now to use an example, we will look back at our table.

let us look for the conditional probability of:

$p_(X|Y)(3|4) = 0.1/0.4 = 0.25$

Thus, the probability that $X = 3$ given that $Y=4$ is $0.25$