The problem asks to find two probabilities involving marbles drawn from an urn:

1. The probability that all three marbles have the same color.
2. The probability that none of the marbles are white.

Information provided:

• 7 red marbles
• 10 white marbles
• 10 blue marbles
• Total marbles = $7 + 10 + 10 = 27$
• Three marbles are selected without replacement.

1. Probability all three marbles have the same color:

We will consider the cases where the selected marbles are either all red, all white, or all blue. Since this is without replacement, the probabilities will be calculated for each color separately, and then summed.

• All three red: The number of ways to choose 3 red marbles from 7 is $\binom{7}{3}$. The total number of ways to choose 3 marbles from 27 is $\binom{27}{3}$. $\text{P(all red)} = \frac{\binom{7}{3}}{\binom{27}{3}} = \frac{\frac{7!}{3!(7-3)!}}{\frac{27!}{3!(27-3)!}} = \frac{35}{2925}$

• All three white: $\text{P(all white)} = \frac{\binom{10}{3}}{\binom{27}{3}} = \frac{120}{2925}$

• All three blue: $\text{P(all blue)} = \frac{\binom{10}{3}}{\binom{27}{3}} = \frac{120}{2925}$

Now, sum these probabilities to get the total probability of drawing three marbles of the same color: $\text{Total probability} = \frac{35}{2925} + \frac{120}{2925} + \frac{120}{2925} = \frac{275}{2925} \approx 0.094$

2. Probability that none of the marbles are white:

If none of the marbles are white, all three must come from the red and blue marbles only (7 red + 10 blue = 17 marbles total).

• The number of ways to choose 3 marbles from the 17 red and blue marbles is $\binom{17}{3}$. $\text{P(no white)} = \frac{\binom{17}{3}}{\binom{27}{3}} = \frac{680}{2925} \approx 0.2325$

Would you like further details or explanations on any of the steps?

Additional Questions:

1. What is the probability of selecting at least one red marble?
2. What is the probability of selecting exactly one white marble?
3. How would the probability change if the selection was with replacement?
4. What is the probability that exactly two marbles are the same color?
5. How does the probability change if two marbles are selected instead of three?

Tip: When dealing with probabilities, it's important to consider all possible outcomes for a more accurate result.