Question

156 paving stones were examined for cracks, and 12 were found to be cracked. The same 156 stones were examined for discoloration, and 16 were found to be discolored. A total of 3 stones were both cracked and discolored. What is the probability of a stone being either cracked or discolored?

Answer 1

The probability of a stone being either cracked or discolored is 25/156, which is approximately 0.1603.

Step-by-step explanation:

To find the probability of a stone being either cracked or discolored, we need to use the principle of inclusion-exclusion. Let's start by finding the probability of a stone being cracked, denoted as P(C). We know that out of the 156 stones examined, 12 were found to be cracked. So, P(C) = 12/156. Similarly, let's find the probability of a stone being discolored, denoted as P(D). We know that out of the 156 stones examined, 16 were found to be discolored. So, P(D) = 16/156.

Now, we need to find the probability of a stone being both cracked and discolored, denoted as P(C ∩ D). We know that 3 stones were found to be both cracked and discolored. So, P(C ∩ D) = 3/156.

Using the principle of inclusion-exclusion, the probability of a stone being either cracked or discolored is equal to the sum of individual probabilities minus the probability of their intersection:

P(C ∪ D) = P(C) + P(D) - P(C ∩ D)

Now, we can substitute the values we have:

P(C ∪ D) = (12/156) + (16/156) - (3/156) = 25/156 ≈ 0.1603