Question

A bag contains 5 red marbles, 10 white marbles, and 9 blue marbles. You draw 3 marbles out at random, without replacement. What is the probability that exactly two of the marbles are red? What is the probability that none of the marbles are red?

Answer 1

To calculate the probability of drawing exactly two red marbles out of three without replacement, we'll use the concept of combinations.

Total number of marbles in the bag: 5 red + 10 white + 9 blue = 24 marbles

Probability of drawing exactly two red marbles:
Step 1: Calculate the total number of ways to draw 3 marbles out of 24 (without considering colors). This can be calculated using combinations:

C(24, 3) = 24! / (3! * (24 - 3)!) = 24! / (3! * 21!) = (24 * 23 * 22) / (3 * 2 * 1) = 2024

Step 2: Calculate the number of ways to draw exactly two red marbles. This can be calculated using combinations as well:

C(5, 2) = 5! / (2! * (5 - 2)!) = 5! / (2! * 3!) = (5 * 4) / (2 * 1) = 10

Step 3: Calculate the number of ways to draw one non-red marble (white or blue) out of the remaining marbles:

C(10+9, 1) = C(19, 1) = 19

Step 4: Calculate the probability of drawing exactly two red marbles:

Probability = (Number of ways to draw exactly two red marbles) * (Number of ways to draw one non-red marble) / (Total number of ways to draw 3 marbles)

Probability = (10 * 19) / 2024 ≈ 0.0935 or 9.35%

Now, let's calculate the probability of drawing none of the marbles being red:

Probability of drawing none of the marbles being red:
Step 1: Calculate the total number of ways to draw 3 non-red marbles out of 19 (10 white + 9 blue) without considering colors:

C(10+9, 3) = C(19, 3) = 19! / (3! * (19 - 3)!) = 19! / (3! * 16!) = (19 * 18 * 17) / (3 * 2 * 1) = 969

Step 2: Calculate the total number of ways to draw 3 marbles out of 24 (considering all marbles):

C(24, 3) = 2024 (as calculated before)

Step 3: Calculate the probability of drawing none of the marbles being red:

Probability = (Number of ways to draw 3 non-red marbles) / (Total number of ways to draw 3 marbles)

Probability = 969 / 2024 ≈ 0.4789 or 47.89%

So, the probability of drawing exactly two red marbles is approximately 9.35%, and the probability of drawing none of the marbles being red is approximately 47.89%.

Answer 2

..

Explanation:

p (r r w) × order and p ( r r b) × order

=5/24× 4/23 × 10/22 ×3!/2! + 5/24× 4/23 × 9/22 ×3!/2!

=95/1012

none red:

p ( r bar, r bar, r bar)

=19/24 × 18/23 × 17/22 × 3! = 2907/1012