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 calculate the probabilities for each scenario based on the given situation. 

You have:
- 4 blue balls
- 3 white balls
- Total balls = 4 (blue) + 3 (white) = 7 balls

### 1. Probability of picking two blue balls:

You are drawing two balls without replacement, so this will involve combinations.

The probability of picking two blue balls is given by the ratio of the favorable outcomes to the total possible outcomes.

#### Total possible outcomes:
This is the number of ways to choose 2 balls from 7 balls:
\[
\binom{7}{2} = \frac{7 \times 6}{2} = 21
\]

#### Favorable outcomes (two blue balls):
This is the number of ways to choose 2 blue balls from 4 blue balls:
\[
\binom{4}{2} = \frac{4 \times 3}{2} = 6
\]

Thus, the probability of drawing two blue balls is:
\[
P(\text{2 blue}) = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{6}{21} = \frac{2}{7}
\]

### 2. Probability of picking one blue and one white ball:

There are two possible orders: you could pick a blue first, then a white, or you could pick a white first, then a blue. However, both will give the same result in terms of probability.

#### Favorable outcomes (1 blue and 1 white):
The number of ways to choose 1 blue ball from 4 blue balls is:
\[
\binom{4}{1} = 4
\]

The number of ways to choose 1 white ball from 3 white balls is:
\[
\binom{3}{1} = 3
\]

Thus, the number of favorable outcomes for picking one blue and one white is:
\[
4 \times 3 = 12
\]

So, the probability of drawing one blue and one white is:
\[
P(\text{1 blue, 1 white}) = \frac{12}{21} = \frac{4}{7}
\]

### Final Answer:
- Probability of picking two blue balls: \( \frac{2}{7} \)
- Probability of picking one blue and one white ball: \( \frac{4}{7} \)

Would you like further details on any of these steps? Here are some related questions you might find helpful:

1. What if there were 5 blue balls and 3 white balls instead?
2. How does the probability change if I pick three balls instead of two?
3. What is the probability of picking two white balls?
4. How does the probability change if I replace one blue ball after picking?
5. How does the probability change if I add more blue balls to the total?

**Tip**: In probability problems involving combinations, always think about whether order matters—this determines if you need combinations or permutations.

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

Calculate the probability of drawing two blue balls or one blue and one white ball from a set of 4 blue and 3 white balls. This problem explores fundamental concepts of probability and combinatorics, using combination formulas and probability calculations. Suitable for students in Grades 8-10.

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