# bag with $60$ balls, count how many are white

Suppose we have a bag with $$60$$ balls of the same size. Their colors are red, black and white. To pick a red ball with absolute certainty, we have to pick at least $$36$$ balls. For a black one, we have to pick at least $$40$$. How many white balls are there in the bag?

I am not sure what the task means by 'absolute certainty'. How could I count the total number of white balls since we are talking about probabilities here?

• there are exactly 35 balls that are not red.
– Surb
Aug 29 '20 at 10:22
• and so exactly 19 that are not black, hence there are 15 that are white? Aug 29 '20 at 10:23
• No, 39 that are not black.
– Surb
Aug 29 '20 at 10:25

## 2 Answers

With absolute certainty, means that if you randomly pick 36 balls out of 60 balls, every time, there will be at least one red ball. This implies that there are exactly 35 balls of color which is not red. This implies that there are exactly 25 balls which are red in color. Using same logic for black balls, we get that there are exactly 21 black balls. Hence 14 white balls present in the bag of 60 balls.

Hint: If $$b,w,r$$ denote the number of black, white and red balls respectively, then you get \begin{align} b+w=35\\ r+w=39\\ r+b+w=60 \end{align}