题目
Florence Hixon is screening a set of 100 stocks based on two criteria (Criterion 1 and Criterion 2). She set the passing level such that 50% of the stocks passed each screen. For these stocks, the values for Criterion 1 and Criterion 2 are not independent but are positively related. How many stocks should pass Hixon’s two screens?
选项
A.Less than 25
B.25
C.More than 25
答案
C
解析
C is correct. Let event A be a stock passing the first screen (Criterion 1) and event B be a stock passing the second screen (Criterion 2). The probability of passing each screen is P(A) = 0.50 and P(B) = 0.50. If the two criteria are independent, the joint probability of passing both screens is P(AB) = P(A)P(B) = 0.50 × 0.50 = 0.25, so 25 out of 100 stocks would pass both screens. However, the two criteria are positively related, and P(AB) ≠ 0.25. Using the multiplication rule for probabilities, the joint probability of A and B is P(AB) = P(A | B) P(B). If the two criteria are not independent, and if P(B) = 0.50, then the contingent probability of P(A | B) is greater than 0.50. So the joint probability of P(AB) = P(A | B)P(B) is greater than 0.25. More than 25 stocks should pass the two screens. C正确。假设一只股票通过标准1的事件为A，一只股票通过标准2的事件为B，那么通过两个标准的概率=事件AB同时发生的概率=P(AB)=P(A)*P(B/A) 因为标准1和标准2的设定是正相关的，那也就意味着当股票通过其中一个标准时，再通过另一个标准的概率是增加的，故P(B/A)>P(B)=0.5 所以P(AB)=P(A)*P(B/A)>0.25，因此最后通过两个标准的股票数量是大于25的