To find the probability of picking out 5 white and 3 red balls from a bag containing 12 white and 3 red balls, you need to divide the number of favorable outcomes by the number of possible outcomes.
First, let's calculate the number of favorable outcomes. In this case, the favorable outcome is selecting 5 white balls from the initial 12 white balls and selecting all 3 red balls.
To calculate the number of ways to select 5 white balls from 12, you can use the combination formula, denoted as C(n,r), where n represents the total number of items and r represents the number of items you want to select. The formula is:
C(n,r) = n! / (r! * (n-r)!)
In this case, the number of ways to select 5 white balls from 12 is:
C(12,5) = 12! / (5! * (12-5)!)
= 12! / (5! * 7!)
= (12 * 11 * 10 * 9 * 8) / (5 * 4 * 3 * 2 * 1)
= 792
Next, since we want to select all 3 red balls, there is only one possible way to do so.
Therefore, the number of favorable outcomes is 792 * 1 = 792.
Now, let's calculate the number of possible outcomes, which is the total number of ways to select 8 balls (5 white + 3 red) from the initial 15 balls (12 white + 3 red). Again, using the combination formula:
C(15,8) = 15! / (8! * (15-8)!)
= 15! / (8! * 7!)
= (15 * 14 * 13 * 12 * 11 * 10 * 9) / (8 * 7 * 6 * 5 * 4 * 3 * 2 * 1)
= 6435
Therefore, the number of possible outcomes is 6435.
To find the probability, divide the number of favorable outcomes by the number of possible outcomes:
Probability = Number of Favorable Outcomes / Number of Possible Outcomes
= 792 / 6435
≈ 0.1231
So, the probability of picking out 5 white and 3 red balls from the given bag is approximately 0.1231, or about 12.31%.