Câu hỏi:

Let A be the event that the bulb is đạt chuẩn (meets the standard). Let \overline{A} be the event that the bulb is not đạt chuẩn. Let B be the event that the bulb is not rejected by the device S. We are given: P(A) = 0.95, P(\overline{A}) = 0.05. If a bulb is not đạt chuẩn, S rejects it with probability 0.99. So, P(reject | \overline{A}) = 0.99, which means P(not reject | \overline{A}) = P(B | \overline{A}) = 1 - 0.99 = 0.01. 10% of the rejected bulbs are actually đạt chuẩn. So, P(đạt chuẩn | reject) = 0.1. This means P(reject | A) = 0.1. We want to find P(A | B) = P(A | not rejected) = $\frac{P(B|A)P(A)}{P(B)}$ P(B|A) = 1 - P(reject | A) = 1 - 0.1 = 0.9. P(B) = P(B|A)P(A) + P(B|\overline{A})P(\overline{A}) = (0.9)(0.95) + (0.01)(0.05) = 0.855 + 0.0005 = 0.8555. So, P(A|B) = $\frac{(0.9)(0.95)}{0.8555} = \frac{0.855}{0.8555} = \frac{8550}{8555} = \frac{1710}{1711}$. Therefore, a = 1710 and b = 1711. We need to calculate a+b = 1710+1711 = 3421. However, the answer choices don't have 3421. I seem to make calculation mistake. We are looking for P(A|not rejected)=P(A|B). P(B|A) = 1-0.1 = 0.9. P(B|A') = 1-0.99=0.01. P(A)=0.95, P(A')=0.05. P(A|B) = P(B|A)*P(A) / (P(B|A)*P(A) + P(B|A')*P(A')) = 0.9*0.95 / (0.9*0.95 + 0.01*0.05)=0.855 / (0.855+0.0005)=0.855 / 0.8555=1710/1711. a+b= 3421, I still get this answer. Check if i need to find P(A'| not rejected). There must be something wrong. Let B be the event the bulb is rejected. Then the question means finding P(A|B'). Using bayes formula. P(A|B')= P(B'|A)*P(A) / (P(B'|A)*P(A) + P(B'|A')*P(A')). Since P(A')=0.05, P(A)=0.95. P(B|A')= 0.99. So, P(B'|A')= 0.01. since 10% of rejected lamps are đạt chuẩn. P(A|B) = 0.1, thus, P(A'|B)=0.9. Since P(B|A)=0.1. So, P(B'|A)=0.9. P(A|B') = 0.9*0.95 / (0.9*0.95+ 0.01*0.05)= 0.855/(0.855+0.0005)= 0.855 /0.8555 = 1710/1711. The result is very close to 1. Maybe wrong. Maybe the correct understanding of the question is the device removes 0.99 chance of those substandard items. 10% of removed ones are ok? P(A|reject) = 0.1 = x, thus P(not a| reject) = 0.9 =y. Then , how can it be solved?