May 14, 2013

After going through the last five years sample papers for the CFA Program Level 1 Exam provided by the CFA Program Institute, I found out that about 10% of the questions relate directly to probability concepts. While the concepts in probability are simple, the error rate in answering such questions is alarming. All of these questions can be done correctly with mere attentiveness and keeping the following points in mind while solving them.

** Tip # 1 : Complete the Sample Set **Let us say we have a game which gives outcomes between 15 and 30. What could be the sample set? Take a shot at this!

About 70% of the people who have been asked this question blindly take the sample set with 15 elements! Think again, and the answer becomes clear: it is a set with 16 elements! And in the exam paper, there would certainly be an option according to that value, which will make you pick the wrong answer.

__Tip # 2__ : **Revise the Probabilities **Another highly common mistake is avoiding the additional qualitative data given within the question and focusing on the numbers instead. A classic example where even mathematicians have gone wrong is the Monty Hall Problem, where unless the role of the host is considered, one can never get the right solution. HHHave u heard of it?? If not, Google it now!

Let us take a simple example: There are four red and three black balls in a bag. Assume you are picking out balls without replacement. What is the probability of picking a black ball in the second trial if red ball was drawn in the first trial?

What some people do is to ignore the fact that the balls are taken out without replacement and take the probability of picking the black ball as 2/6 or 0.33. But this is incorrect! The question states that the balls are picked out with replacement; we need to consider the revised probabilities. Thus, if a red ball is picked out in the first trial, the number of balls available to us is 3 red and 3 black. Hence the probability is 0.5.

__Tip # 3__ : **Don’t get confused between Conditional and Reverse Conditional Probability**

This kind of confusion is most common in questions involving returns on more than one asset, or when the returns on an asset depends upon factors other than chance. Let us understand this through an example.

Jack lives in a society where 3% of the people are drug users. A test can correctly identify users 95% of the time and incorrectly identify non-users 90% of the time. Given that Jack was identified as a drug user, what is the probability that he indeed is a drug user?

In this question, without any information, the probability of Jack using drugs is 3%. But given the test results, we now need to find out the reverse conditional probability.

We have:

The point to note is that the probabilities of being tested positive if one is a drug user (95%), and being a drug user if one is tested positive (0.227) are entirely different. Such question should be carefully dealt with to avoid misleading results.

__Tip # 4__ : **Remember that the Normal Distribution table gives Cumulative Probabilities **

Many a times, questions involving normal distribution requires the use of tables. Students pay utmost attention in finding the value corresponding to the right numbers, but forget to subtract the appropriate probability in order to find out the right answer. And CFA Program Institute traps you again by providing an option corresponding to the values you (mistakenly) calculated.

Let’s take an example: A stock has an average EPS of $6 and a standard deviation of $2. What is the probability of getting a return of less than $7 but more than the mean?

Usually people just find out the probability of getting return below $7, which is 0.69, directly given in the table. But we also need to find out the probability of getting returns below the mean, which is 0.5. Hence, the correct answer is 0.69-0.5 = 0.19.

__Tip # 5__ : **Ignore Bogus Data and use Basic Axioms to eliminate options**

Some of the work can be avoided when one takes some time to study the options. Sometimes, a lot of irrelevant data is provided in the question. Instead of trying to use all of them, identify the main question asked and check if that can be answered using the options. For example, consider the following question:

Alice wants to invest $20000 in the stock market. There are three possible returns: good, average and bad, which depend upon the state of the economy. The state can be either progressive or recessive. The probability of the economy being recessive is 55%. Given the state is progressive; the probability of getting good returns is 74%. Given the economy is recessive; the probability of getting bad returns is 81%. Alice decided to invest $15000 after thorough discussion with her friend Bob. What is the probability that the economy will turn out to be good?

**a. 33.3%**

**b. 45%**

**c. 40.7%**

This question does not require any of the data provided in the question except the fact that the probability of the economy being in recessive state is 55%. The answer is (b) 45%.

Always remember to verify the answer before making the final choice. In case, it does not fit, there is something wrong and you probably need to solve the question in the long way.

Keep the above points in mind when you attempt questions in probability and avoid all silly mistakes that can cost you the CFA Program certification! Stay tuned for the next article on quant series for the exhaustive list of must-know in hypothesis testing!

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