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The problem provides information that 80% of diamonds are classified as industrial-grade. The task is to determine the probability that randomly chosen diamonds are industrial-grade.

Let’s solve both parts (a) and (b):

### Part (a)
Two diamonds are chosen at random. The probability that both diamonds are industrial-grade:

1. The probability that the first diamond is industrial-grade is \(0.80\).
2. The probability that the second diamond is also industrial-grade is \(0.80\).

Since these events are independent, the probability of both diamonds being industrial-grade is:
\[
P(\text{both industrial-grade}) = 0.80 \times 0.80 = 0.64
\]

### Part (b)
Four diamonds are chosen at random. The probability that all four diamonds are industrial-grade:

1. The probability that the first diamond is industrial-grade is \(0.80\).
2. The probability for each subsequent diamond being industrial-grade is also \(0.80\).

So, the probability of all four diamonds being industrial-grade is:
\[
P(\text{all four industrial-grade}) = 0.80 \times 0.80 \times 0.80 \times 0.80 = 0.80^4 = 0.4096
\]

### Final Answers:
- (a) \( P(\text{two diamonds}) = 0.6400 \)
- (b) \( P(\text{four diamonds}) = 0.4096 \)

Would you like further details or clarifications on any part?

Here are 5 related questions for practice:
1. What is the probability that none of the two diamonds chosen are industrial-grade?
2. If one diamond is chosen, what is the probability that it is gem-quality?
3. What is the probability that at least one of two diamonds chosen is industrial-grade?
4. If three diamonds are chosen, what is the probability that exactly two are industrial-grade?
5. How would the probabilities change if only 70% of diamonds were industrial-grade?

**Tip:** When dealing with independent events, the probability of multiple events happening together is found by multiplying the individual probabilities of each event.

A diamond can be classified as either gem-quality or industrial-grade. Suppose that 80% of diamonds are classified as industrial-grade. (a) Two diamonds are chosen at random. What is the probability that both diamonds are industrial-grade? (b) Four diamonds are chosen at random. What is the probability that all four diamonds are industrial-grade?

Learn how to calculate the probability of selecting industrial-grade diamonds in random samples, assuming 80% of diamonds are industrial-grade. The solution covers independent events and probability concepts suitable for Grades 9-12.

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