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?
it's easy to fin P1(no cubes) or P2(no spheres).
P(some of each) = 1 - (P1+P2)
Could you explain in more detail? I don't understand
To calculate the probability of getting at least one cube and one sphere when randomly picking 5 objects without replacement from the bag, we need to consider the different possibilities.
1. Calculate the total number of ways to choose 5 objects from the bag without replacement:
The total number of objects in the bag is 11 cubes + 16 spheres = 27 objects.
The total number of ways to choose 5 objects from 27 is denoted as C(27, 5) or 27 choose 5, which is calculated as 27! / (5! * (27 - 5)!) = 27! / (5! * 22!).
2. Calculate the number of ways to choose at least one cube and one sphere:
We can consider two cases - one with one cube and four spheres, and another with two cubes and three spheres.
Case 1: One cube and four spheres
- Number of ways to choose 1 cube from 11 cubes is C(11, 1) = 11
- Number of ways to choose 4 spheres from 16 spheres is C(16, 4) = 1820
- Number of ways to choose 1 cube and 4 spheres is 11 * 1820 = 20020
Case 2: Two cubes and three spheres
- Number of ways to choose 2 cubes from 11 cubes is C(11, 2) = 55
- Number of ways to choose 3 spheres from 16 spheres is C(16, 3) = 560
- Number of ways to choose 2 cubes and 3 spheres is 55 * 560 = 30800
3. Calculate the total number of ways to pick at least one cube and one sphere:
Total Number of ways = Number of ways in Case 1 + Number of ways in Case 2
Total Number of ways = 20020 + 30800 = 50820
4. Finally, calculate the probability:
Probability = (Total Number of ways) / (Total Number of ways to choose 5 objects from the bag without replacement)
Probability = 50820 / C(27, 5)
You can use a calculator or any programming language to calculate the combination formula C(n, r). In this case, C(27, 5) is equal to 27! / (5! * 22!).