Questions: An urn contains 5 white and 7 black balls. Four balls are randomly drawn from the urn in succession, with replacement. That is, after each draw, the selected ball is returned to the urn. What is the probability that all 4 balls drawn from the urn are white? Round your answer to three decimal places.

Solution Steps

An urn contains 5 white balls and 7 black balls, totaling 12 balls. Four balls are drawn with replacement, meaning each draw is independent, and the probability of drawing a white ball remains the same for each draw. We need to find the probability that all 4 balls drawn are white.

The probability of drawing a white ball in a single draw is the number of white balls divided by the total number of balls: \[ P(\text{White}) = \frac{5}{12}. \]

Since the draws are independent and with replacement, the probability of drawing 4 white balls in succession is the product of the probabilities of drawing a white ball in each draw: \[ P(\text{All 4 White}) = \left(\frac{5}{12}\right)^4. \]

Calculate \(\left(\frac{5}{12}\right)^4\): \[ \left(\frac{5}{12}\right)^4 = \frac{625}{20736} \approx 0.0301. \]

Final Answer