麻烦老师详细解释下这道题目的含义吧，而且答案计算的逻辑也没看明白

问题如下：

Question

• P(FCF exceeded consensus) = 0.50

• P(FCF met consensus) = 0.35

• P(FCF fell short of consensus) = 0.15

While waiting for Abco to release last quarter’s FCF data, the investor learns that Abco will acquire a competitor. Believing that the upcoming acquisition makes it more likely that last quarter’s FCF will exceed the consensus, the investor generates a list of FCF events that may have influenced the acquisition:

• P(Acquisition | FCF exceeded consensus) = 0.40

• P(Acquisition | FCF met consensus) = 0.25

• P(Acquisition | FCF fell short of consensus) = 0.35

Using Bayes’ Formula, calculate the probability that Abco is likely to exceed consensus FCF expectations for last quarter given the acquisition. P(FCF exceeded consensus | Acquisition) is closest to:

An investor in Abco stock forecasts the probability that Abco exceeded, met, or fell short of consensus expectations for free cash flow (FCF) during the prior quarter:

选项：

A.34%. B.59%. C.27%.

解释：

Solution

B is correct. The updated probability P(FCF exceeded consensus | Acquisition) is 59%.

1. Calculate the unconditional probability that Abco will acquire the competitor firm:P(Acquisition) = (0.50 × 0.40) + (0.35 × 0.25) + (0.15 × 0.35) = 0.34, or 34%.

2. Calculate the updated probability that Abco exceeded consensus expectations for FCF given that they acquire the competitor firm: P(FCF exceeded consensus | Acquisition) = [P(Acquisition | FCF exceeded consensus)/P(Acquisition)] × P(FCF exceeded consensus) = (0.40/0.34) × (0.50) = 0.59 or 59%.

A is incorrect because 34% is the unconditional probability that Abco acquires the competitor firm: P(Abco acquires) = (0.50 × 0.40) + (0.35 × 0.25) + (0.15 × 0.35) = 0.34, or 34%.

C is incorrect because the updated probability was calculated in error:

[P(Acquisition | FCF exceeded consensus) × P(Acquisition)]/P(FCF exceeded consensus) = (0.40 × 0.34)/(0.50) = 0.27 or 27%