Question

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. Suppose that the probability of drawing each ball is the same. What is the probability that both drawn balls are black?

Hint:

In problems involving drawing without replacement, always adjust the total number of outcomes after each draw.

Answer 1

Step 1: Understanding the situation.
There are 15 balls in total, 10 black and 5 white. We need to find the probability that both balls drawn are black, considering that the draws are without replacement.

Step 2: Finding the probability of drawing the first black ball.
The probability of drawing a black ball on the first draw is: \[ P(\text{1st black}) = \frac{10}{15} = \frac{2}{3}. \]

Step 3: Finding the probability of drawing the second black ball.
After drawing the first black ball, there are now 9 black balls left and 14 balls in total. The probability of drawing a black ball on the second draw is: \[ P(\text{2nd black}) = \frac{9}{14}. \]

Step 4: Calculating the overall probability.
Since the draws are without replacement, the total probability is the product of the individual probabilities: \[ P(\text{both black}) = \frac{2}{3} \times \frac{9}{14} = \frac{18}{42} = \frac{3}{7}. \]

Step 5: Conclusion.
Thus, the probability that both drawn balls are black is \( \frac{3}{7} \).