To solve this problem, we need to calculate the probability that the first ball drawn is blue and the second ball drawn is red, given that the balls are drawn without replacement.

Step-by-step Solution:

1. Total number of balls:
   The jar contains 5 blue balls and 3 red balls, so the total number of balls is: $5 \text{ (blue balls)} + 3 \text{ (red balls)} = 8 \text{ balls in total}.$

2. Probability of drawing a blue ball first:
   There are 5 blue balls out of a total of 8 balls. So, the probability of drawing a blue ball first is: $P(\text{First ball is blue}) = \frac{5}{8}.$

3. Probability of drawing a red ball second (after a blue ball is drawn):
   After drawing one blue ball, there are 7 balls left in the jar (since we are drawing without replacement). Out of the remaining balls, 3 are red. So, the probability of drawing a red ball second is: $P(\text{Second ball is red | First ball is blue}) = \frac{3}{7}.$

4. Finding the total probability:
   To find the total probability of both events happening (first blue, second red), we multiply the probabilities of the individual events: $P(\text{First blue, second red}) = \frac{5}{8} \times \frac{3}{7} = \frac{15}{56}.$

Thus, the probability that the first ball drawn is blue and the second ball drawn is red is $\frac{15}{56}$.

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Would you like more details on how to approach similar problems or any questions about this solution?

Here are some related questions to explore:

1. What is the probability that both balls drawn are of the same color (both blue or both red)?
2. How does the probability change if the balls were drawn with replacement?
3. How can you calculate the probability of drawing at least one red ball in two draws?
4. What would the probabilities be if there were more red balls than blue balls in the jar?
5. How would you approach a similar problem with more than two balls drawn?

Tip: When drawing without replacement, the total number of outcomes decreases with each draw, so make sure to adjust the denominators accordingly for each subsequent draw.