A bag contains 10 marbles: 2 are green, 5 are red, and 3 are blue. Miguel chooses a marble at random, and without putting it back, chooses another one at random. What is the probability that both marbles he chooses are red? Write your answer as a fraction in simplest form.

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).

Learn how to calculate the probability of drawing two red marbles consecutively from a bag containing 10 marbles (2 green, 5 red, and 3 blue). This step-by-step solution demonstrates the use of probability for dependent events. Ideal for students in Grades 6-8.

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