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# "quantity"exercise: joint probability

Questions 1:

Event X and Event Y are independent events. The probability of X is 0.2 [P(X) = 0.2] and the probability of Y is 0.5 [P(Y) = 0.5]. The joint probability of X and Y, P(XY), is closest to:

A、 0.7.

B、 0.3.

C、 0.1.

Questions 2:

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:

● 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 investor 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:

A、 34%.

B、 59%.

C、 27%.

【analysis】

C is correct. Given that X and Y are independent, their joint probability is equal to the product of their individual probabilities. In this case: P(XY) = P(X)P(Y) = 0.2 × 0.5 = 0.1.

A is incorrect; it is the sum of the two individual probabilities (i.e., 0.5 + 0.2 = 0.7).

B is incorrect; it is the difference between the two individual probabilities (i.e., 0.5 – 0.2 = 0.3).

【analysis】

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%

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