An urn contains 8 blue marbles and 7 red marbles. A sample of 6 marbles is chosen from the urn without replacement. What is the probability that the sample contains at least one blue marble?
what is P(all red)?
Subtract that from 1.
To find the probability that the sample contains at least one blue marble, we first need to determine the total number of possible outcomes and the number of favorable outcomes.
The total number of possible outcomes can be calculated using combinations. We have a total of 15 marbles in the urn, and we are choosing a sample of 6 marbles. So, the total number of possible outcomes is given by:
C(15, 6) = 15! / (6! * (15 - 6)!) = 15! / (6! * 9!)
Next, we need to determine the number of favorable outcomes, which is the number of ways to choose at least one blue marble. There are two scenarios to consider:
Scenario 1: Choosing exactly one blue marble and the remaining 5 marbles from the remaining 14 marbles.
In this case, we have 8 ways to choose one blue marble and C(14, 5) ways to choose the remaining 5 marbles. Therefore, the number of favorable outcomes for scenario 1 is: 8 * C(14, 5)
Scenario 2: Choosing more than one blue marble. This means we choose 2 or more blue marbles and the remaining marbles from the remaining 14 marbles.
In this case, we can calculate the number of favorable outcomes by summing up the number of ways to choose exactly 2, 3, 4, 5, and 6 blue marbles. Therefore, the number of favorable outcomes for scenario 2 is: C(8, 2) * C(14, 4) + C(8, 3) * C(14, 3) + C(8, 4) * C(14, 2) + C(8, 5) * C(14, 1) + C(8, 6)
Finally, we can calculate the probability by dividing the number of favorable outcomes by the total number of possible outcomes:
P(at least one blue marble) = (Number of favorable outcomes) / (Total number of possible outcomes) = (Number of favorable outcomes for scenario 1 + Number of favorable outcomes for scenario 2) / (Total number of possible outcomes)
Evaluate the expressions for the number of favorable outcomes and the total number of possible outcomes and divide them to find the probability.