If I have 4 blue balls and 3 white balls, what is a propability, if I pick two blue balls, and what is propability, if I pick one blue and one white ball?

Let's solve this step by step:

### 1. Probability of Picking Two Blue Balls:
You have a total of 7 balls (4 blue and 3 white). The probability of picking two blue balls can be found by calculating the following:

- **First pick**: The probability of picking a blue ball is \(\frac{4}{7}\) (since there are 4 blue balls out of 7 total balls).
- **Second pick**: After removing one blue ball, there are now 6 balls left, and 3 of them are blue. So, the probability of picking a second blue ball is \(\frac{3}{6}\).

The probability of both events happening (picking two blue balls) is the product of these two probabilities:

\[
P(\text{Two blue balls}) = \frac{4}{7} \times \frac{3}{6} = \frac{4}{7} \times \frac{1}{2} = \frac{4}{14} = \frac{2}{7}
\]

So, the probability of picking two blue balls is **\(\frac{2}{7}\)**.

### 2. Probability of Picking One Blue and One White Ball:
There are two ways to pick one blue and one white ball:

- **First blue, then white**:
  - First pick: Probability of picking a blue ball = \(\frac{4}{7}\).
  - Second pick: After removing one blue ball, there are 6 balls left, 3 of which are white. So, the probability of picking a white ball is \(\frac{3}{6} = \frac{1}{2}\).

  Probability for this sequence:
  \[
  P(\text{Blue, then white}) = \frac{4}{7} \times \frac{1}{2} = \frac{4}{14} = \frac{2}{7}
  \]

- **First white, then blue**:
  - First pick: Probability of picking a white ball = \(\frac{3}{7}\).
  - Second pick: After removing one white ball, there are 6 balls left, and 4 of them are blue. So, the probability of picking a blue ball is \(\frac{4}{6} = \frac{2}{3}\).

  Probability for this sequence:
  \[
  P(\text{White, then blue}) = \frac{3}{7} \times \frac{2}{3} = \frac{6}{21} = \frac{2}{7}
  \]

The total probability of picking one blue and one white ball is the sum of these two probabilities:

\[
P(\text{One blue, one white}) = \frac{2}{7} + \frac{2}{7} = \frac{4}{7}
\]

### Summary:
- The probability of picking two blue balls is **\(\frac{2}{7}\)**.
- The probability of picking one blue and one white ball is **\(\frac{4}{7}\)**.

Would you like further details or have any questions?

Here are 5 related questions to explore further:
1. What is the probability of picking two white balls?
2. What is the probability of picking no blue balls at all?
3. How would the probabilities change if we added more blue or white balls?
4. How does the probability change if we put back the first ball after each pick?
5. Can you explain how the combinations formula would be used here?

**Tip:** Always check if the drawing is with or without replacement, as it affects the probabilities!

If I have 4 blue balls and 3 white balls, what is the probability if I pick two blue balls, and what is the probability if I pick one blue and one white ball?

Discover how to calculate the probability of picking two blue balls or one blue and one white ball from a set of 4 blue and 3 white balls. Learn step-by-step how to apply basic probability and combinatorics concepts, suitable for Grades 6-8 students.

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