To calculate the combined probability of drawing two different colored balls without replacement, you need to multiply the probability of drawing each ball in sequence.

Consider a scenario where you have a bag containing red and blue balls. The first step is to determine the probability of drawing one color, followed by the probability of drawing the other color, given that the first ball has already been removed.

For instance, suppose you have $3$ red balls and $2$ blue balls in a bag, making the total number of balls $5$. The probability of drawing a red ball first is given by:

After removing a red ball, there are now $4$ balls left in the bag (which consist of $2$ red balls and $2$ blue balls). The probability of subsequently drawing a blue ball is:

To find the joint probability of these two events occurring in sequence, you multiply the individual probabilities:

Next, consider the reverse scenario: drawing a blue ball first and then a red ball. The probability of drawing a blue ball first is:

After drawing a blue ball, there are still $4$ balls left (this time consisting of $3$ red balls and $1$ blue ball). The probability of then drawing a red ball is:

Again, we find the joint probability of these two events occurring in sequence:

Finally, to find the overall probability of drawing two different colored balls without replacement, we add the probabilities from both scenarios:

Thus, the combined probability of drawing two different colored balls without replacement is $\frac{3}{5}$.