An urn contains 5 white and 6 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 black? Round your answer to three decimal places.

Question

An urn contains 5 white and 6 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 black? Round your answer to three decimal places.

Answer 1

To calculate the probability of drawing all 4 black balls, we need to determine the probability of drawing one black ball in each of the four successive draws.

Given that there are 11 balls in total (5 white and 6 black), the probability of drawing a black ball in one draw is the ratio of the number of black balls to the total number of balls. So, the probability of drawing one black ball is:

P(drawing one black ball) = Number of black balls / Total number of balls

P(drawing one black ball) = 6 / 11

Since we are drawing with replacement, the probability of drawing a black ball in each of the four draws is independent. So, we can calculate the probability of drawing all four black balls by multiplying the individual probabilities together:

P(drawing all 4 black balls) = P(drawing one black ball) * P(drawing one black ball) * P(drawing one black ball) * P(drawing one black ball)

P(drawing all 4 black balls) = (6 / 11) * (6 / 11) * (6 / 11) * (6 / 11)

Now, let's calculate the exact value:

P(drawing all 4 black balls) ≈ 0.086 (rounded to three decimal places)

Therefore, the probability of drawing all 4 black balls from the urn is approximately 0.086 or 8.6% (rounded).

Answer 2

To find the probability of drawing 4 black balls in succession with replacement, we need to calculate the probability of drawing a black ball in each individual draw and then multiply these probabilities together.

The probability of drawing a black ball in each draw is given by the ratio of the number of black balls to the total number of balls in the urn:

P(drawing a black ball) = 6/11

Since each draw is independent and we are replacing the drawn ball after each draw, we can multiply the probabilities together to find the probability of getting 4 black balls in a row:

P(drawing 4 black balls in a row) = (6/11) * (6/11) * (6/11) * (6/11)

Simplifying this expression:

P(drawing 4 black balls in a row) = 1296/14641

Rounding to three decimal places:

P(drawing 4 black balls in a row) ≈ 0.088

Answer 3

since the draws are independent, and

P(black) = 6/11

P(4 blacks) = (6/11)^4