An urn contains 7 pink and 8 black balls. Five balls are randomly drawn from the urn in succession, with replacement. What is the probability that all five balls drawn from the urn are black?
prob = C(8,5)/C(15,5) = 56/3003 = 8/429
or
prob = (8/15)(7/14)(6/13)(5/12)(4/11)
= 8/429
To find the probability that all five balls drawn from the urn are black, we need to determine the probability of drawing a black ball on each individual draw and then calculate the product of these probabilities.
Step 1: Calculate the probability of drawing a black ball on the first draw.
There are a total of 7 pink and 8 black balls, so the probability of drawing a black ball on the first draw is 8/15.
Step 2: Since the balls are replaced after each draw, the probability of drawing a black ball on the second draw is also 8/15.
Step 3: Similarly, the probability of drawing a black ball on the third, fourth, and fifth draws is 8/15.
Step 4: Calculate the product of these probabilities.
Since the events are independent, we can multiply the probabilities. Therefore, the probability of drawing five black balls in succession is (8/15) * (8/15) * (8/15) * (8/15) * (8/15).
Step 5: Simplify the product.
Calculating this product gives us (8/15)^5.
Step 6: Calculate the final probability.
Evaluating (8/15)^5 gives us approximately 0.068.
Therefore, the probability that all five balls drawn from the urn are black is approximately 0.068, or 6.8%.
To find the probability that all five balls drawn from the urn are black, we need to first determine the probability of drawing a black ball on each individual draw.
The total number of balls in the urn is 7 pink + 8 black = 15 balls. Since the balls are drawn with replacement, the probability of drawing a black ball on each draw will remain constant.
The probability of drawing a black ball on any given draw is calculated by dividing the number of black balls by the total number of balls in the urn.
Therefore, the probability of drawing a black ball on each draw is 8 black balls / 15 total balls = 8/15.
Since the balls are drawn with replacement, the probability of each draw is independent of the others. Therefore, to find the probability of all five balls being black, we need to multiply the probabilities of each individual draw.
So, the probability of drawing five black balls in succession is (8/15) * (8/15) * (8/15) * (8/15) * (8/15) = (8/15)^5 = 0.0590 (rounded to four decimal places).
Therefore, the probability that all five balls drawn from the urn are black is approximately 0.0590 or 5.90%.