Suppose a jar contains 19 red marbles and 19 blue marbles. If you reach in the jar and pull out 2 marbles at random at the same time, find the probability that both are red.

Enter your answer using at least 4 decimal places.

P(both red)

To find the probability that both marbles drawn are red, we can approach this problem using the concept of combinations.

### Step 1: Calculate the total number of ways to choose 2 marbles from the jar
The total number of marbles in the jar is \( 19 + 19 = 38 \). The number of ways to choose 2 marbles from 38 is given by the combination formula:

\[
\text{Total combinations} = \binom{38}{2} = \frac{38 \times 37}{2} = 703
\]

### Step 2: Calculate the number of ways to choose 2 red marbles
There are 19 red marbles, so the number of ways to choose 2 red marbles is:

\[
\text{Red combinations} = \binom{19}{2} = \frac{19 \times 18}{2} = 171
\]

### Step 3: Calculate the probability
The probability that both marbles drawn are red is the ratio of the number of favorable outcomes (choosing 2 red marbles) to the total number of outcomes (choosing any 2 marbles):

\[
P(\text{both red}) = \frac{\text{Red combinations}}{\text{Total combinations}} = \frac{171}{703} \approx 0.2433
\]

Thus, the probability that both marbles are red is approximately **0.2433**.

Would you like more details or have any questions?

Here are 5 related questions:
1. What is the probability of drawing 2 blue marbles?
2. How would the probability change if there were 20 red marbles and 18 blue marbles?
3. What is the probability of drawing one red marble and one blue marble?
4. If 3 marbles were drawn, what is the probability that all are red?
5. How does the probability change if the marbles are drawn one after the other without replacement?

**Tip:** When calculating probabilities involving combinations, it's crucial to carefully count both the total and favorable outcomes to ensure accuracy.

Learn how to calculate the probability of drawing two red marbles from a jar containing 19 red marbles and 19 blue marbles. This problem involves using combinations to determine favorable outcomes and total outcomes, essential in probability calculations. Suitable for advanced high school students.

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