Calculating the probability of error free products using Beyes theorem

Question

My Problem

When producing a product, two errors (A and B) can occur.
The probability that a product has error A : P(A) = 0.05
The probability that the product has error B, given that it has error A : P(B|A) = 0.4
The probability that a product has error B, given that it does not have error A : P(B|A*) = 0.15
What is the probability that a product is completely free from errors?

Answer:
$0.81$

My attempt of solving it
$P(\text{error free}) = 1-P(B|A) = 0.6$

My Question:
I am clearly getting the wrong answer.
I am thankful for any help and/or guidance.

Answer 1

Hint:

• What is the probability a product does not have error A?
• Given that it does not have error A, what is the probability it does not have error B?
• What is the probability it does not have error A and it does not have error B?