Question

There are 18 balls in a crate. 5 are glass, 7 are plastic, 4 are ceramic, 2 are metal. a. How many ways can you choose 3 balls? b. If 3 balls are randomly chosen, what's the probability they're all glass?

Answer 1

There are 816 ways to choose 3 balls from 18, and the probability that all three balls chosen are glass is approximately 0.0041.

Step-by-step explanation:

To answer the student's question, we need to calculate the number of ways to choose 3 balls from 18, and then determine the probability that all three balls chosen are glass.

Part a: Number of Ways to Choose 3 Balls

The number of ways to choose 3 balls from 18 without regard to order can be determined by using the combination formula, which is:

C(n, k) = n! / (k!(n-k)!)

For 18 balls, where n=18 and we are choosing k=3 balls, the calculation is:

C(18, 3) = 18! / (3!(18-3)!) = (18 × 17 × 16) / (3 × 2 × 1) = 816

Part b: Probability of Choosing All Glass Balls

To find the probability of choosing all glass balls when three balls are chosen at random, we use the formula for the probability of A and B and C:

P(All Glass) = P(Glass1) × P(Glass2 | Glass1) × P(Glass3 | Glass1 and Glass2)

Since there are 5 glass balls and 18 total balls:

P(Glass1) = 5/18

After one glass ball is selected, there are now 4 glass balls left and 17 total balls:

P(Glass2 | Glass1) = 4/17

After selecting a second glass ball, there are now 3 glass balls left and 16 total balls:

P(Glass3 | Glass1 and Glass2) = 3/16

Now, multiply these probabilities together:

P(All Glass) = (5/18) × (4/17) × (3/16) = 60 / (18 × 17 × 16) = 0.0041 (approximately)