Question

Ryan has a pack of 10 pens. 3 pens are red, 4 are black, and 3 are blue. What are the chances that he pulls out a red pen and a black pen? Assume that the pens are pulled without replacement. Round to the nearest whole number.(1 poin

Answer 1

To calculate the chances of Ryan pulling out a red pen and a black pen from a pack of 10 pens without replacement, we can break the problem down into two scenarios: first, he pulls out a red pen and then he pulls out a black pen, and second, he pulls out a black pen followed by a red pen.

### Scenario 1: Red pen first, then black pen
1. The probability of picking a red pen first:
\[
P(\text{Red first}) = \frac{3}{10}
\]
2. After pulling out a red pen, there are 9 pens left, including 4 black pens. The probability of now picking a black pen:
\[
P(\text{Black second | Red first}) = \frac{4}{9}
\]
3. Therefore, the combined probability for this scenario:
\[
P(\text{Red first, Black second}) = P(\text{Red first}) \times P(\text{Black second | Red first}) = \frac{3}{10} \times \frac{4}{9} = \frac{12}{90}
\]

### Scenario 2: Black pen first, then red pen
1. The probability of picking a black pen first:
\[
P(\text{Black first}) = \frac{4}{10}
\]
2. After pulling out a black pen, there are still 3 red pens among the remaining 9 pens. The probability of now picking a red pen:
\[
P(\text{Red second | Black first}) = \frac{3}{9}
\]
3. Therefore, the combined probability for this scenario:
\[
P(\text{Black first, Red second}) = P(\text{Black first}) \times P(\text{Red second | Black first}) = \frac{4}{10} \times \frac{3}{9} = \frac{12}{90}
\]

### Total Probability
Now, we can add the two probabilities together:
\[
P(\text{Red and Black}) = P(\text{Red first, Black second}) + P(\text{Black first, Red second}) = \frac{12}{90} + \frac{12}{90} = \frac{24}{90}
\]
This simplifies to:
\[
P(\text{Red and Black}) = \frac{4}{15}
\]

### Convert to Percentage
To find the percentage, we multiply by 100:
\[
\frac{4}{15} \approx 0.2667
\]
So as a percentage:
\[
0.2667 \times 100 \approx 26.67\%
\]
Rounding to the nearest whole number, the chances are:
\[
\boxed{27\%}
\]