A bag contains 33 balls: 10 blue, 11 red, and 12 green. If a person will select balls at random from the bag without replacement, what is the minimum number of balls that need to be selected in order to be sure that the balls selected include 3 balls of the same color?
This is more of a logic question. Let B = blue, R = red, and G = green. If you choose 3 balls, they COULD be three of the same color, but not necessarily. They could be B,R,G. Four balls could be B,B,R,G. Five balls could be B,B,R,R,G. Six balls could be B,B,R,R,G,G and you would still not have three of the same color. However, if you pick one more ball (seven balls total), you are assured that at least three will be of one color. The answer is seven.
To be sure that the balls selected include 3 balls of the same color, we need to consider the worst-case scenario for each color.
1. Firstly, let's consider the worst-case scenario for blue balls. In this case, we need to select all 10 blue balls, which ensures that we have at least 3 blue balls.
2. Secondly, let's consider the worst-case scenario for red balls. After selecting all 10 blue balls, there are 23 balls left in the bag, out of which 11 are red balls. To guarantee that we have at least 3 red balls, we need to select an additional 3 red balls.
3. Finally, let's consider the worst-case scenario for green balls. After selecting the 10 blue balls and 3 red balls, there are 20 balls left in the bag, out of which 12 are green balls. To ensure that we have at least 3 green balls, we need to select an additional 3 green balls.
Therefore, in order to be sure that the balls selected include 3 balls of the same color, the minimum number of balls that need to be selected is:
10 blue balls + 3 red balls + 3 green balls = 16 balls.
So, a minimum of 16 balls need to be selected.
To determine the minimum number of balls that need to be selected to ensure that you have at least 3 balls of the same color, we can consider the worst-case scenario.
In this case, the worst-case scenario would be selecting all the balls of one or two colors except for 2. Let's analyze two scenarios:
1. Selecting all the blue and red balls: In this scenario, you would need to select 10 blue balls and 11 red balls to ensure that you have at least 3 balls of the same color. This makes a total of 21 balls.
2. Selecting all the red and green balls: Similarly, you would need to select 11 red balls and 12 green balls to ensure that you have at least 3 balls of the same color, totaling 23 balls.
Since we are looking for the minimum number of balls, we take the larger value among the two scenarios. Therefore, you would need to select a minimum of 23 balls to guarantee that you have at least 3 balls of the same color.