A jar contains 20 marbles: 4 red, 6 white and 10 blue. If you remove 1 marble at a time, randomly, what is the minimum number that you must remove to be certain that you have at least 2 marble of each color?
I guessed the answer was 18 and it was correct but I don't know how to solve it. Please help!
If you remove 16, you could possibly have 10 blue and 6 white. You must remove two more in order to be certain that you have at least two of each color.
Well, if you want to be certain that you have at least 2 marbles of each color, there's a rule of thumb you can follow. You should remove the maximum number of marbles of the color that appears the least. In this case, red is the least common color with only 4 marbles. So you should remove all the red marbles first.
Next, you want to make sure you have at least 2 of each color left. Since you've already removed 4 red marbles, there are 16 marbles left. You can now focus on removing the next most common color, which is white. You can remove a maximum of 4 white marbles, leaving you with 12 marbles.
Finally, you'll want to remove the remaining 2 blue marbles in order to ensure you have at least 2 of each color. So, you need to remove a total of 4 red marbles, 4 white marbles, and 2 blue marbles. That adds up to 10 marbles.
Therefore, the minimum number of marbles you must remove to be certain that you have at least 2 marbles of each color is 10.
To solve this problem, we can consider the worst-case scenario. In order to have at least 2 marbles of each color, we need to ensure that we have not removed both of any color before getting the 2nd marble of that color.
Starting with the most abundant color, we want to make sure we have not removed all 10 blue marbles before getting the 2nd marble of another color. Therefore, we need to remove at least 10 marbles.
Next, we consider the next most abundant color, which is white with 6 marbles. We need to make sure we have not removed all 6 white marbles before getting the 2nd marble of another color. Therefore, we need to remove at least 6 more marbles.
Finally, we consider the least abundant color, which is red with 4 marbles. We need to make sure we have not removed all 4 red marbles before getting the 2nd marble of another color. Therefore, we need to remove at least 4 more marbles.
Adding up the minimum number of marbles to remove from each color, we have: 10 + 6 + 4 = 20.
Therefore, you must remove at least 20 marbles to be certain that you have at least 2 marbles of each color.
To solve this problem, we need to consider the worst-case scenario. We want to find the minimum number of marbles we need to remove in order to be certain that we have at least 2 marbles of each color.
In the worst-case scenario, we would have removed all marbles of one or two colors before getting at least 2 marbles of the remaining color(s). So, let's consider the two extreme cases:
1. Assume we remove all the red and white marbles first. In this case, we need to remove all the red marbles (4) and all the white marbles (6) before we start removing the blue marbles.
2. Assume we remove all the red and blue marbles first. In this case, we need to remove all the red marbles (4) and all the blue marbles (10) before we start removing the white marbles.
To ensure we have at least 2 marbles of each color, we need to remove the maximum number of marbles from either of these two scenarios. So, we take the larger number from each scenario, which gives us 6 (removing all red marbles) and 10 (removing all blue marbles).
Now, if we add these two numbers together, we get 6 + 10 = 16. However, this number does not account for the fact that we have not yet removed any white marbles. So, we need to remove at least 2 white marbles as well.
Therefore, the minimum number of marbles we must remove to be certain that we have at least 2 marbles of each color is 16 (removing all red and blue marbles) + 2 (removing at least 2 white marbles) = 18.
So, your initial guess of 18 marbles being the minimum number to be certain of having 2 marbles of each color is indeed correct.