Question

There are 7 black balls and 10 red balls in an urn. If 3 balls are drawn without replacement, what is the probability that more than 1 black ball is drawn? Express your answer as a fraction or a decim

Answer 1

To calculate the probability of drawing more than one black ball from an urn with 7 black balls and 10 red balls when drawing 3 balls without replacement, one must add the probability of drawing exactly two black balls and one red ball to the probability of drawing all three black balls.

Step-by-step explanation:

The question asks for the probability of drawing more than one black ball when three balls are drawn without replacement from an urn containing 7 black balls and 10 red balls. To find this probability, we need to consider the different ways to draw more than one black ball, which could be exactly two black balls and one red ball, or all three black balls.

• The probability of drawing two black balls and one red ball can be calculated as follows: P(2 Black, 1 Red) = (7/17) * (6/16) * (10/15) + (7/17) * (10/16) * (6/15) + (10/17) * (7/16) * (6/15) = 7 * 6 * 10 / 17 * 16 * 15. This accounts for all possible orders of drawing two black balls and one red ball.
• The probability of drawing three black balls is P(3 Black) = (7/17) * (6/16) * (5/15).

Add these probabilities together to get the total probability of drawing more than one black ball. Let's denote this probability as P(More than 1 Black). Therefore, P(More than 1 Black) = P(2 Black, 1 Red) + P(3 Black).

Answer 2

the probability that more than 1 black ball is drawn when 3 balls are drawn without replacement from the urn is approximately 0.7133 or 71.33%.

To find the probability that more than 1 black ball is drawn when 3 balls are drawn without replacement from an urn containing 7 black balls and 10 red balls, we can use the principle of complementary probability. We'll calculate the probability of the complement event (i.e., the event where 0 or 1 black ball is drawn) and then subtract it from 1 to get the probability of more than 1 black ball being drawn.

Let's break it down step by step:

Step 1: Calculate the probability of drawing 0 black balls and 3 red balls:

- Probability of drawing a red ball on the first draw = (10 red balls) / (17 total balls) = 10/17

- Probability of drawing a red ball on the second draw (without replacement) = (9 red balls remaining) / (16 total balls remaining) = 9/16

- Probability of drawing a red ball on the third draw (without replacement) = (8 red balls remaining) / (15 total balls remaining) = 8/15

Now, multiply these probabilities together because they are independent events:

Probability of drawing 0 black balls and 3 red balls = (10/17) * (9/16) * (8/15)

Step 2: Calculate the probability of drawing 1 black ball and 2 red balls:

- Probability of drawing a black ball on the first draw = (7 black balls) / (17 total balls) = 7/17

- Probability of drawing a red ball on the second draw (without replacement) = (10 red balls) / (16 total balls remaining) = 10/16 (simplify to 5/8)

- Probability of drawing a red ball on the third draw (without replacement) = (9 red balls remaining) / (15 total balls remaining) = 9/15 (simplify to 3/5)

Now, multiply these probabilities together because they are independent events:

Probability of drawing 1 black ball and 2 red balls = (7/17) * (5/8) * (3/5)

Step 3: Calculate the complement probability (0 or 1 black ball drawn):

Add the probabilities from Step 1 and Step 2 to get the probability of drawing 0 or 1 black ball:

Complement Probability = Probability of drawing 0 black balls and 3 red balls + Probability of drawing 1 black ball and 2 red balls

Now, add the two probabilities together:

Complement Probability = [(10/17) * (9/16) * (8/15)] + [(7/17) * (5/8) * (3/5)]

Step 4: Calculate the probability of more than 1 black ball drawn (complement of the complement):

Now that we have the complement probability, subtract it from 1 to find the probability of more than 1 black ball being drawn:

Probability of more than 1 black ball drawn = 1 - Complement Probability

calculate the probabilities step by step:

Step 1: Calculate the probability of drawing 0 black balls and 3 red balls:

Probability of drawing 0 black balls and 3 red balls = (10/17) * (9/16) * (8/15)

Probability of drawing 0 black balls and 3 red balls ≈ 0.1396 (rounded to four decimal places)

Step 2: Calculate the probability of drawing 1 black ball and 2 red balls:

Probability of drawing 1 black ball and 2 red balls = (7/17) * (5/8) * (3/5)

Probability of drawing 1 black ball and 2 red balls ≈ 0.1471 (rounded to four decimal places)

Step 3: Calculate the complement probability (0 or 1 black ball drawn):

Complement Probability = Probability of drawing 0 black balls and 3 red balls + Probability of drawing 1 black ball and 2 red balls

Complement Probability ≈ 0.1396 + 0.1471 ≈ 0.2867 (rounded to four decimal places)

Step 4: Calculate the probability of more than 1 black ball drawn (complement of the complement):

Probability of more than 1 black ball drawn = 1 - Complement Probability

Probability of more than 1 black ball drawn = 1 - 0.2867 ≈ 0.7133 (rounded to four decimal places)

So, the probability that more than 1 black ball is drawn when 3 balls are drawn without replacement from the urn is approximately 0.7133 or 71.33%.