she does this total of 30 times and selects 6 pens and 10 blue pens and 14 black pens

Question

based on this information what is the probaility that cynara will select blue pen return it to the bag and then select red pen?

Answer 1

To find the probability that Cynara will select a blue pen, return it to the bag, and then select a red pen, we need to first clarify how many pens of each color are present and the total number of pens.

From your information:

• Total selections: 30
• Blue pens selected: 10
• Black pens selected: 14
• Total pens (selected): 6 + 10 + 14 = 30 pens

Let's assume all the pens are of equal distribution, as you haven't provided the total number of each colored pen. Therefore, Cynara has:

• 0 red pens mentioned
• 6 unspecified colored pens

Assuming that the 6 pens selected were not blue or black, we will have:

1. Selecting a Blue Pen: The probability of selecting a blue pen depends on the total number of blue pens in the bag. We know 10 blue pens were selected out of 30 total selections, but without specifics on how many blue pens are originally in the bag, we cannot find an exact probability.

2. Returning the Blue Pen: The condition states she returns the blue pen to the bag, meaning the total number of pens remains unchanged after this event.

3. Selecting a Red Pen: Since you mention red pens but have not indicated how many are there, logically, if it hasn’t been mentioned, we would assume there are none, which leads to an impossible event of selecting a red pen.

If we assume there is at least one red pen in a total of 30 pens and using the data shared, the overall probability setup can be based on hypothetical values knowing exactly how many red pens exist.

Hypothetical Case with Values:

If in total we assumed:

• 10 Blue Pens
• 14 Black Pens
• 6 Other Colored Pens (let's assume they are Red for this calculation)

The probability calculation with added assumptions:

1. Total Pens = 10 (Blue) + 14 (Black) + 6 (Red) = 30 Pens
2. Probability of selecting a Blue Pen: \[ P(\text{Blue}) = \frac{\text{Number of Blue Pens}}{\text{Total Pens}} = \frac{10}{30} = \frac{1}{3} \]
3. She returns it to the bag and repeats:
4. Probability of selecting a Red Pen afterwards: \[ P(\text{Red}) = \frac{\text{Number of Red Pens}}{\text{Total Pens}} = \frac{6}{30} = \frac{1}{5} \]

Combined Probability:

\[ P(\text{Blue then Red}) = P(\text{Blue}) \times P(\text{Red}) = \frac{1}{3} \times \frac{1}{5} = \frac{1}{15} \]

Conclusion:

For an accurate probability assessment, we would need more information about the number of red pens present. If there are no red pens, the probability is 0. If there are red pens, the above combination applies. Please clarify or provide the number of red pens to refine the calculation!