Question

A company manufactures marbles. Over a year, it is observed that 70% of the marbles manufactured pass the quality check and 30% are rejected. 60% of the rejected marbles undergo reprocessing, while the rest go into scrap. What is the probability that out of 15 marbles selected at random exactly 3 marbles go into scrap?

Answer 1

Probability that exactly 3 marbles are sent to scrap is 0.1696

Explanation:

The percentage of marbles of the initial production that is rejected is 30%. On the other hand, of that 30% 40% is sent to scrap metal. Where the percentage of marbles sent to scrap is (30%) (40%) = 12%. In the same way, the probability that any marble will be sent to scrap is 0.12.

Now, the number of marbles that should be sent to scrap, within a group of 15 marbles, is a binomial random variable with parameters n = 15 and p = 0.12. Therefore, the probability that exactly 3 marbles are sent to scrap is:

`P(X = 3) = _(n)C_(x) (p)^x(1-p)^(n-x) = _(15)C_(3) (0.12)^3(0.88)^(12) = 0.1696`

Answer 2

The probability that 1 given marble goes into scrap is: (30% chance of rejection)*(40% change of scrap if rejected) = 0.3*0.4 = 0.12. Therefore the probability that a marble does not go into scrap is 1 - 0.12 = 0.88.

Based on a binomial distribution:
P = (15C3)(0.12)^3 (0.88)^(15-3) = 0.1696.