there are 30 marbles in a bag; 4 blue, 10 red,6 white, 10 yellow. You randomly draw5 marbles without replacement. Find the percent of drawing 2 red marbles and at least 2 white marbles.
To find the percent of drawing 2 red marbles and at least 2 white marbles, we need to determine the total number of favorable outcomes and the total number of possible outcomes.
First, let's find the total number of ways to choose 5 marbles out of 30. This can be calculated using the combination formula, also known as "n choose r":
nCr = n! / (r!(n-r)!)
In our case, n = 30 (total number of marbles) and r = 5 (number of marbles drawn). Let's calculate it:
30C5 = 30! / (5!(30-5)!)
= 30! / (5! * 25!)
= (30 * 29 * 28 * 27 * 26) / (5 * 4 * 3 * 2 * 1)
= 142,506
So, the total number of possible outcomes is 142,506.
Now, let's find the total number of ways to choose 2 red marbles and at least 2 white marbles.
To find the number of ways to choose 2 red marbles, we use the combination formula again:
10C2 = 10! / (2!(10-2)!)
= 10! / (2! * 8!)
= (10 * 9) / (2 * 1)
= 45
To find the number of ways to choose at least 2 white marbles, we consider two cases:
Case 1: Choosing exactly 2 white marbles
To find the number of ways to choose 2 white marbles, we use the combination formula:
6C2 = 6! / (2!(6-2)!)
= 6! / (2! * 4!)
= (6 * 5) / (2 * 1)
= 15
Case 2: Choosing 3, 4, or 5 white marbles
We subtract the number of ways to choose exactly 1 or no white marbles from the total number of possible outcomes.
Number of ways to choose exactly 1 white marble:
6C1 = 6
Number of ways to choose no white marble:
24C5 = 24! / (5!(24-5)!)
= 24! / (5! * 19!)
= (24 * 23 * 22 * 21 * 20) / (5 * 4 * 3 * 2 * 1)
= 42,504
So, the number of ways to choose at least 2 white marbles is:
Total - (Number of ways to choose exactly 1 white marble + Number of ways to choose no white marble)
= 142,506 - (6 + 42,504)
= 99,996
Now, let's find the total number of favorable outcomes, which is the product of the number of ways to choose 2 red marbles and at least 2 white marbles:
Total favorable outcomes = (Number of ways to choose 2 red marbles) * (Number of ways to choose at least 2 white marbles)
= 45 * 99,996
= 4,499,820
Finally, to calculate the percentage, we divide the total favorable outcomes by the total possible outcomes and multiply by 100:
Percentage = (Total favorable outcomes / Total possible outcomes) * 100
= (4,499,820 / 142,506) * 100
≈ 3.16%
Therefore, the approximate percent of drawing 2 red marbles and at least 2 white marbles is approximately 3.16%.