Problem:
A bag contains 3 black and 4 red balls. Two balls are drawn at random, one at a time without replacement. What is the probability?

Solution:

To find the probability, we need to determine the number of favorable outcomes and the total number of possible outcomes.

Favorable Outcomes:
There are two cases where we can have favorable outcomes:
1. Drawing a black ball and then a red ball.
2. Drawing a red ball and then a black ball.

Favorable outcome 1:
In the first case, we have 3 black balls and 4 red balls. The probability of drawing a black ball on the first draw is 3/7 (since there are 7 balls in total). After drawing a black ball, there are 6 balls remaining in the bag, out of which 4 are red. So, the probability of drawing a red ball on the second draw is 4/6. Therefore, the probability of this favorable outcome is (3/7) * (4/6) = 12/42 = 2/7.

Favorable outcome 2:
In the second case, we have 3 black balls and 4 red balls. The probability of drawing a red ball on the first draw is 4/7 (since there are 7 balls in total). After drawing a red ball, there are 6 balls remaining in the bag, out of which 3 are black. So, the probability of drawing a black ball on the second draw is 3/6. Therefore, the probability of this favorable outcome is (4/7) * (3/6) = 12/42 = 2/7.

Total Possible Outcomes:
To find the total number of possible outcomes, we need to consider all the ways in which we can draw two balls from the bag without replacement.

The first ball can be any of the 7 balls in the bag. After drawing the first ball, there are 6 balls remaining in the bag. So, the second ball can be any of the 6 remaining balls. Therefore, the total number of possible outcomes is 7 * 6 = 42.

Probability:
Now, we can find the probability by dividing the number of favorable outcomes by the total number of possible outcomes.

Probability = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
Probability = (2/7) / (42/42)
Probability = 2/7

Therefore, the probability of drawing one black ball and one red ball from the bag is 2/7.