A batch of glass bricks are to be inspected for defects. The probability of having air bubbles inside a brick is 0.004. If there are air bubbles inside a brick, the probability that it will be cracked is 0.6. If there are no ahubbles inside a brick, the probability that it will be cracked is 0.005. If a brick is chosen randomly, what is thprobability that it will be cracked?
Question
A batch of glass bricks are to be inspected for defects. The probability of having air bubbles inside a brick is 0.004. If there are air bubbles inside a brick, the probability that it will be cracked is 0.6. If there are no ahubbles inside a brick, the probability that it will be cracked is 0.005. If a brick is chosen randomly, what is thprobability that it will be cracked?
Solution
To solve this problem, we need to use the law of total probability. The law of total probability states that the probability of an event can be found by considering all the different ways that it can happen.
Step 1: Identify the probabilities given in the problem.
- The probability of having air bubbles inside a brick (P(B)) is 0.004.
- The probability that a brick with air bubbles will be cracked (P(C|B)) is 0.6.
- The probability that a brick without air bubbles will be cracked (P(C|~B)) is 0.005.
Step 2: Calculate the probability of not having air bubbles inside a brick (P(~B)).
P(~B) = 1 - P(B) = 1 - 0.004 = 0.996.
Step 3: Use the law of total probability to find the probability that a brick will be cracked (P(C)).
P(C) = P(B) * P(C|B) + P(~B) * P(C|~B) = 0.004 * 0.6 + 0.996 * 0.005 = 0.0024 + 0.00498 = 0.00738.
So, the probability that a brick will be cracked is 0.00738 or 0.738%.
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