I have a bag with 2 different kinds of objects: 17 cubes, 19 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 one sphere when randomly picking 5 objects from the bag without replacement, we need to consider two cases:

Case 1: Picking 1 cube and 4 spheres
Case 2: Picking 1 sphere and 4 cubes

Let's calculate the probability for each case and then add the two probabilities together.

Case 1: Picking 1 cube and 4 spheres

To calculate the probability of picking 1 cube and 4 spheres, we need to calculate the probability for each event and multiply them together:

Probability of picking 1 cube: 17/36 (since there are 17 cubes out of a total of 36 objects)
Probability of picking 4 spheres: 19/35, 18/34, 17/33, 16/32

To calculate the probability of picking 4 spheres, we divide the number of spheres left in the bag by the total number of remaining objects each time we pick one.

So, the probability for Case 1 is (17/36) * (19/35) * (18/34) * (17/33) * (16/32).

Case 2: Picking 1 sphere and 4 cubes

The probability for Case 2 is:
Probability of picking 1 sphere: 19/36
Probability of picking 4 cubes: 17/35, 16/34, 15/33, 14/32

Again, we divide the number of cubes left in the bag by the total number of remaining objects each time we pick one.

So, the probability for Case 2 is (19/36) * (17/35) * (16/34) * (15/33) * (14/32).

Finally, we add the probabilities for Case 1 and Case 2 to get the total probability of getting at least one cube and one sphere:
(17/36) * (19/35) * (18/34) * (17/33) * (16/32) + (19/36) * (17/35) * (16/34) * (15/33) * (14/32).

By calculating this expression, we can get the probability of getting at least one of each kind of object when randomly picking 5 objects.