in a bag there are 20 red 30black 40 blue50 white balls. what is the minimum number of balls to be drawn without replacement so that you are certsin about getting 4 red,5 black,6 blue,7 white balls
Red is the hardest to get, since there are only 20 of them.
To make sure you get some red, the worst case is you picked all the black, blue and whites (totalling 120) and still not get a red.
What is the minimum you need?
To determine the minimum number of balls you need to draw without replacement to be certain about getting certain quantities of each color, you can use the concept of the worst-case scenario. In this scenario, you assume that every ball you draw is of a color that is different from the one you need, until only the desired colors remain.
In this case, you want to be certain about getting 4 red, 5 black, 6 blue, and 7 white balls. So let's calculate the worst-case scenario for each color individually:
For red balls:
Since you need a minimum of 4 red balls, the worst-case scenario would be to draw all other balls except the red ones first. Therefore, you would need to draw all 30 black balls, 40 blue balls, and 50 white balls before you start drawing red balls. So far, you would have drawn a total of 30 + 40 + 50 = 120 balls.
To be certain about getting at least 4 red balls, you need to draw all the remaining red balls, which is 20.
For black balls:
Since you need a minimum of 5 black balls, you should consider the worst-case scenario again. Assuming all the red balls have already been drawn, you would need to draw all remaining blue balls (40) and white balls (50) first. So far, you would have drawn a total of 40 + 50 = 90 balls.
To be certain about getting at least 5 black balls, you need to draw all the remaining black balls, which is 30.
For blue balls:
Using the same logic, assuming all the red and black balls have been drawn, you would need to draw all the remaining white balls (50) first, which brings the total number of balls drawn to 120 (from drawing red balls) + 90 (from drawing black balls) + 50 (from drawing white balls) = 260.
To be certain about getting at least 6 blue balls, you need to draw all the remaining blue balls, which is 40.
For white balls:
Again, assuming all the red, black, and blue balls have been drawn, you need to draw all the remaining white balls to be certain about getting at least 7, which is 50.
Now, let's sum up the total number of balls drawn in each worst-case scenario:
120 (for red) + 90 (for black) + 260 (for blue) + 50 (for white) = 520
Therefore, the minimum number of balls you need to draw without replacement to be certain about getting 4 red, 5 black, 6 blue, and 7 white balls is 520.