Cheryl Smith, CFA, is comparing dividend changes for energy and non-energy companies. Smith determines that 15% of the stock market universe consists of energy companies. Smith also determines that the probability that an energy company will increase its dividend is 90% and the probability that a non-energy company will increase its dividend is 30%. If Smith randomly selects one company from the universe of stocks and notices that the company declared a dividend increase, the probability that the company Smith selected is an energy company is closest to:
A) 5%.
B) 15%.
C) 35%.
Explanation
C is correct
Initially, we are told that the probability of randomly selecting an energy company is 15% (“Smith determines that 15% of the stock market universe consists of energy companies”). This is the prior probability. But Smith gathers new information that she can use to update her prior probability—the randomly selected stock recently declared a dividend increase. Intuitively, we know the updated probability will be much higher than 15% because dividend increases are likely to occur much more often for energy companies (90% of the time) than for non-energy companies (30% of the time). Thus, the correct choice must be 35%. We can obtain the exact answer using Bayes’ formula:
where:
- P(E) is the prior probability that the selected company is an energy company
- P(D|E) is the probability of the new information (dividend increase) for a given event (energy company).
- P(D) is the unconditional probability of the new information (dividend increase).
P(E) is provided in the question as 15%. P(D|E) also is provided in the question as 90%. P(D) is the probability of a dividend increase, which can occur either if the randomly selected company is an energy company or a non-energy company:
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