Let's calculate the probability that the marbles drawn from both boxes are of the same color.

Step 1: Define the events

• Box A contains 5 red marbles and 3 white marbles.
• Box B contains 2 red marbles and 6 white marbles.

The possible outcomes for drawing marbles from both boxes are:

1. Both marbles are red.
2. Both marbles are white.

We need to find the probability of these events and then sum them up.

Step 2: Calculate probabilities for each case

Case 1: Both marbles are red

• Probability of drawing a red marble from Box A:
  $P(\text{Red from A}) = \frac{5}{5 + 3} = \frac{5}{8}$
• Probability of drawing a red marble from Box B:
  $P(\text{Red from B}) = \frac{2}{2 + 6} = \frac{2}{8} = \frac{1}{4}$
• Probability that both marbles are red (independent events):
  $P(\text{Both red}) = P(\text{Red from A}) \times P(\text{Red from B}) = \frac{5}{8} \times \frac{1}{4} = \frac{5}{32}$

Case 2: Both marbles are white

• Probability of drawing a white marble from Box A:
  $P(\text{White from A}) = \frac{3}{5 + 3} = \frac{3}{8}$
• Probability of drawing a white marble from Box B:
  $P(\text{White from B}) = \frac{6}{2 + 6} = \frac{6}{8} = \frac{3}{4}$
• Probability that both marbles are white (independent events):
  $P(\text{Both white}) = P(\text{White from A}) \times P(\text{White from B}) = \frac{3}{8} \times \frac{3}{4} = \frac{9}{32}$

Step 3: Total probability

The total probability that both marbles are of the same color (either both red or both white) is the sum of the two probabilities: $P(\text{Same color}) = P(\text{Both red}) + P(\text{Both white}) = \frac{5}{32} + \frac{9}{32} = \frac{14}{32} = \frac{7}{16}$

Final Answer:

The probability that the marbles drawn from both boxes are of the same color is $\frac{7}{16}$.

---

Would you like further clarification or more examples on this? Here are some related questions:

1. What is the probability of drawing one red and one white marble from both boxes?
2. How does the probability change if the number of red marbles in Box A is increased?
3. What would the probability be if the marbles were drawn without replacement from both boxes?
4. How do the probabilities change if Box B has an equal number of red and white marbles?
5. Can the probability of drawing marbles of the same color be greater than $\frac{1}{2}$?

Tip: When calculating probabilities for independent events, simply multiply the individual probabilities of each event occurring.