I have a bag with 2 different kinds of objects: 11 cubes, 16 spheres. If I randomly pick from the bag 5 of these objects without return any of it, what is the probability of getting at least 1 of each kind?
To find the probability of getting at least one cube and at least one sphere when randomly picking 5 objects, we can use the concept of complementary probability.
First, let's calculate the total number of possible outcomes when picking 5 objects from the bag. Since we have a total of 11 cubes and 16 spheres, the total number of objects in the bag is 11 + 16 = 27. Therefore, the number of ways to choose 5 objects out of these 27 is given by the binomial coefficient, also known as "n choose k":
C(27, 5) = 27! / (5! * (27-5)!) = 27! / (5! * 22!) = (27 * 26 * 25 * 24 * 23) / (5 * 4 * 3 * 2 * 1) = 65,780.
Next, let's calculate the number of outcomes where we don't pick at least one cube or one sphere. This occurs when we only select cubes or only select spheres:
Number of outcomes with only cubes: C(11, 5) = 11! / (5! * (11-5)!) = (11 * 10 * 9 * 8 * 7) / (5 * 4 * 3 * 2 * 1) = 2,772.
Number of outcomes with only spheres: C(16, 5) = 16! / (5! * (16-5)!) = (16 * 15 * 14 * 13 * 12) / (5 * 4 * 3 * 2 * 1) = 4368.
Now, let's calculate the number of outcomes where we pick at least one cube and one sphere. This is the complement of the previous cases:
Number of outcomes with at least one cube and one sphere = Total number of outcomes - (Number of outcomes with only cubes + Number of outcomes with only spheres)
= 65,780 - (2,772 + 4,368)
= 58,640.
Finally, we can compute the probability by dividing the number of successful outcomes (getting at least one cube and one sphere) by the total number of possible outcomes:
Probability of getting at least one cube and one sphere = Number of outcomes with at least one cube and one sphere / Total number of outcomes
= 58,640 / 65,780
= 0.891.
Therefore, the probability of randomly selecting at least one cube and at least one sphere when picking 5 objects from the bag is 0.891, or approximately 89.1%.