The probability of picking one black ball and one red ball from a bag containing 5 black balls and 3 red balls without replacement can be calculated as follows:

There are two possible ways to pick one black ball and one red ball: either we pick a black ball first and then a red ball, or we pick a red ball first and then a black ball.

The probability of picking a black ball first is 5/8, since there are 5 black balls out of a total of 8 balls. After picking a black ball, there are 7 balls left in the bag, of which 3 are red. So, the probability of picking a red ball second is 3/7. Therefore, the probability of picking a black ball first and then a red ball is (5/8) (3/7) = 15/56.

The probability of picking a red ball first is 3/8, since there are 3 red balls out of a total of 8 balls. After picking a red ball, there are 7 balls left in the bag, of which 5 are black. So, the probability of picking a black ball second is 5/7. Therefore, the probability of picking a red ball first and then a black ball is (3/8) (5/7) = 15/56.

Since these two events are mutually exclusive, the probability of either one happening is the sum of their probabilities. So, the probability of picking one black ball and one red ball is (15/56) + (15/56) = 15/28