A bag contains 12 white and 3 red marbles. What is the probability of selecting 8 marbles that must include the 3 reds - that is 5 white and 3 red?
To find the probability of selecting 8 marbles that must include the 3 reds (5 white and 3 red), we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of marbles = 12 white + 3 red = 15 marbles
Total number of ways to select 8 marbles out of 15 marbles = (15 choose 8)
To calculate (15 choose 8), we can use the formula for binomial coefficients:
(15 choose 8) = 15! / (8! * (15-8)!)
Where "!" denotes a factorial, which is the product of all positive integers less than or equal to the given number. For example, 5! = 5 * 4 * 3 * 2 * 1 = 120.
Now, let's calculate the factorial values:
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1 = 40,320
(15-8)! = 7! = 7 * 6 * 5 * 4 * 3 * 2 * 1 = 5,040
So, (15 choose 8) = 15! / (8! * (15-8)!) = 130,750.
Now we need to determine the number of favorable outcomes, which is selecting 5 white and 3 red marbles.
Number of ways to select 5 white marbles out of 12 white marbles = (12 choose 5)
Number of ways to select 3 red marbles out of 3 red marbles = (3 choose 3)
Using the same formula as before, let's calculate these binomial coefficients:
(12 choose 5) = 12! / (5! * (12-5)!) = 792
(3 choose 3) = 3! / (3! * (3-3)!) = 1
So, the number of favorable outcomes = (12 choose 5) * (3 choose 3) = 792 * 1 = 792.
Now, we can calculate the probability by dividing the number of favorable outcomes by the total number of outcomes:
Probability = (Number of favorable outcomes) / (Total number of outcomes)
Probability = 792 / 130,750 ≈ 0.00605
Therefore, the probability of selecting 8 marbles that include 5 white and 3 red is approximately 0.00605, or 0.605%.