To solve this problem, we need to calculate the probability that Miguel chooses two red marbles consecutively without replacement.

Step 1: Total Number of Marbles

Initially, there are 10 marbles in the bag:

• 2 green,
• 5 red,
• 3 blue.

This gives a total of: $10 \text{ marbles}.$

Step 2: Probability of Choosing the First Red Marble

The probability that Miguel selects a red marble on his first draw is the ratio of red marbles to the total number of marbles: $P(\text{first red}) = \frac{5}{10} = \frac{1}{2}.$

Step 3: Probability of Choosing the Second Red Marble

After Miguel chooses a red marble, there are only 9 marbles left in the bag, and 4 of those marbles are red. Therefore, the probability of selecting a second red marble is: $P(\text{second red | first red}) = \frac{4}{9}.$

Step 4: Calculate the Combined Probability

To find the overall probability that both marbles are red, we multiply the probabilities of the two events: $P(\text{both red}) = P(\text{first red}) \times P(\text{second red | first red}) = \frac{1}{2} \times \frac{4}{9} = \frac{4}{18}.$

Step 5: Simplify the Fraction

Now, simplify $\frac{4}{18}$: $\frac{4}{18} = \frac{2}{9}.$

Final Answer:

The probability that both marbles Miguel chooses are red is: $\boxed{\frac{2}{9}}.$

Would you like further details or clarifications?

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Here are 5 related questions you might find useful:

1. What is the probability of selecting one red marble followed by one blue marble?
2. What is the probability of selecting two green marbles in a row?
3. What is the probability that both marbles chosen are of the same color?
4. If Miguel chooses a marble, replaces it, and then chooses again, what is the probability that both are red?
5. How would the probability change if there were 6 red marbles instead of 5?

Tip: When solving probability problems involving multiple events, always check whether the events are dependent (no replacement) or independent (with replacement).