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

1 Answer
May 18, 2015

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#