Probability Calculation for Last Ball Drawn

This problem involves calculating the probability of a specific outcome (the last ball drawn being black) when drawing balls without replacement.

Understanding the Drawing Process

We have a box containing:

• 40 red balls
• 60 black balls
• Total balls = 40 + 60 = 100

Balls are drawn one by one without replacement until the box is empty. We need to find the probability that the very last ball drawn is black.

Key Principle: Symmetry

In a process of drawing items randomly without replacement until none are left, every item has an equal probability of being in any specific position in the sequence of draws. This means any of the 100 balls is equally likely to be the first ball drawn, the second, ..., or the last ball drawn.

Calculating the Probability

Since each of the 100 balls has an equal chance of being the last ball drawn, we can determine the probability based on the composition of the balls:

• Total number of balls = 100
• Number of black balls = 60

The probability that the last ball drawn is black is the ratio of the number of black balls to the total number of balls.

Let B be the event that the last ball drawn is black.

$ P(B) = \frac{\text{Number of black balls}}{\text{Total number of balls}} $

$ P(B) = \frac{60}{100} $

Simplifying the fraction:

$ P(B) = \frac{6}{10} = \frac{3}{5} $

Conclusion

The probability that the last ball drawn is black is $ \frac{3}{5} $.