Question

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

Answer 1

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`

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`