You have a bag of 50 red marbles and 50 blue marbles. Without looking, how many marbles do you need to take out of the bag to guarantee that you have at least 10 marbles of the same color?
Hint: Think carefully about the worst-case scenario.
user
Sure, I can help you with this! Let's first consider the worst-case scenario. The worst-case scenario is when you take out only red marbles or only blue marbles. In that case, you'll have 9 red marbles and 1 blue marble, which is not 10 of the same color.

Now, let's consider another scenario. Suppose you take 11 marbles. Out of these, you can have either 11 red marbles or 11 blue marbles. If you have only 11 red marbles, you've exceeded the minimum of 10 for that color. But, it's unlikely that you'll get 11 blue marbles as well. The number of possible ways to get 11 blue marbles would be (50 choose 11) which is a very small probability (approximately 6.59e-33). Given this, it's unlikely that you would have both a set of 11 red marbles and a set of 11 blue marbles as well. So, taking 11 marbles guarantees you have at least 10 of the same color, which may or may not be a red or blue.

So, taking anything over 10 marbles is redundant. Hence, you would take 10 marbles to guarantee at least one color with 10 marbles.

So, to summarize, you would need 10 marbles to guarantee at least 10 of a single color in the bag.
assistant
You have a bag of 50 red marbles and 50 blue marbles. Without looking, how many marbles do you need to take out of the bag to guarantee that you have at least 10 marbles of the same color?
Hint: Think carefully about the worst-case scenario.
user
In the worst-case scenario, you would draw one marble of each color alternately. So, you could draw 9 red marbles and 9 blue marbles without having 10 of the same color. Therefore, to guarantee that you have at least 10 marbles of the same color, you would need to draw one more marble, for a total of 19 marbles. This ensures that you have at least 10 marbles of the same color, because it's impossible to have more than 9 marbles of one color and still not have 10 of the other color when drawing 19 marbles.

Here's the reasoning:

1. Worst-case scenario: You draw one marble of each color alternately: RBRB...
2. After drawing 9 red and 9 blue marbles, you still don't have 10 of the same color.
3. Draw one more marble:
   a. If it's red, you now have 10 red marbles.
   b. If it's blue, you now have 10 blue marbles.
4. Either way, you now have at least 10 marbles of the same color.

So you need to draw 19 marbles to guarantee having at least 10 marbles of the same color.
assistant