https://file.aiquickdraw.com/mathbot/file/1726400159191a9y0k0z9.jpg

Let's work through the problem step by step.

We have:
- 3 pink pens
- 9 black pens

This gives a total of 12 pens in the pencil case. The pens are chosen at random, **without replacement**, so probabilities will change after the first pen is chosen.

### Part 1: Tree Diagram Probabilities
We want to calculate the probability that:
1. The first pen is black.
2. The second pen is pink.

#### First Pen:
- The probability of picking a **black pen** first is:
  \[
  P(\text{Black first}) = \frac{9}{12} = \frac{3}{4}
  \]

#### Second Pen:
Once a black pen is picked, there are 11 pens left, and the number of pink pens is still 3. The probability of picking a **pink pen** second is:
  \[
  P(\text{Pink second | Black first}) = \frac{3}{11}
  \]

### Total Probability
Since the two events (picking black first, then pink) are independent, we multiply the probabilities:
\[
P(\text{Black first and Pink second}) = \frac{3}{4} \times \frac{3}{11} = \frac{9}{44}
\]

So, the probability that the first pen is black and the second pen is pink is \( \frac{9}{44} \).

Would you like further details or have any questions?

Here are 5 related questions:
1. What is the probability of picking two black pens in a row?
2. How would the probability change if there were 4 pink pens instead of 3?
3. What is the probability that the first pen is pink and the second is black?
4. How would the probabilities change if the pen is returned after the first selection?
5. What is the probability that both pens are the same color?

**Tip**: When dealing with "without replacement" problems, always adjust the total number of items left after each selection.

A pencil case contains 3 pink and 9 black pens. A pen is chosen at random and not returned, and then a second pen is chosen at random. What is the probability that the first pen is black and the second pen is pink?

This math problem involves probability using tree diagrams and conditional probability. Calculate the probability of drawing a black pen first and a pink pen second from a pencil case containing 3 pink and 9 black pens. Suitable for students in Grades 8-10.

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