Two methods, A and B, are available for teaching a certain industrial skill. There is an 80% chance of successfully learning the skill if method A is used, and a 95% chance of success if method B is used. However, method B is substantially more expensive and is therefore used only 25% of the time (method A is used the other 75% of the time). The following notations are suggested:A—Method A is used.B—Method B is used.L—The skill was learned successfully.A worker learned the skill successfully. What is the probability that he was taught by method A?
Question
Two methods, A and B, are available for teaching a certain industrial skill. There is an 80% chance of successfully learning the skill if method A is used, and a 95% chance of success if method B is used. However, method B is substantially more expensive and is therefore used only 25% of the time (method A is used the other 75% of the time). The following notations are suggested:A—Method A is used.B—Method B is used.L—The skill was learned successfully.A worker learned the skill successfully. What is the probability that he was taught by method A?
Solution
To solve this problem, we need to use Bayes' theorem, which is a way to find a probability when we know certain other probabilities. The formula for Bayes' theorem is:
P(A|L) = P(L|A) * P(A) / P(L)
Where:
- P(A|L) is the probability that method A was used given that the skill was learned successfully (this is what we want to find out).
- P(L|A) is the probability of learning the skill successfully given that method A was used (we know this is 0.8).
- P(A) is the probability that method A is used (we know this is 0.75).
- P(L) is the total probability of learning the skill successfully.
First, we need to find P(L), the total probability of learning the skill successfully. We can find this using the law of total probability:
P(L) = P(L and A) + P(L and B) = P(L|A) * P(A) + P(L|B) * P(B) = 0.8 * 0.75 + 0.95 * 0.25 = 0.6 + 0.2375 = 0.8375
Now we can use Bayes' theorem to find P(A|L):
P(A|L) = P(L|A) * P(A) / P(L) = 0.8 * 0.75 / 0.8375 = 0.6 / 0.8375 = 0.716
So, the probability that the worker was taught by method A given that he learned the skill successfully is approximately 0.716 or 71.6%.
Similar Questions
Two methods, A and B, are available for teaching a certain industrial skill. There is an 80% chance of successfully learning the skill if method A is used, and a 95% chance of success if method B is used. However, method B is substantially more expensive and is therefore used only 25% of the time (method A is used the other 75% of the time). The following notations are suggested:A—Method A is used.B—Method B is used.L—The skill was learned successfully.Which of the following is the correct probability tree for this problem?
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