# Level 1 CFA® Exam:

Total Probability Rule

In many cases, we assume that the occurrence of an event is conditional on the occurrence of some possible scenarios. To estimate the probability of such an event we use the total probability rule.

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Note, that two conditions must be met for this formula to hold:

- the sum of event \(S\) and event \(S^C\) should constitute the entire sample space, which means that scenarios \(S\) and \(S^C\) should be exhaustive, and
- events \(S\) and \(S^C\) should not share any elements (so they should be mutually exclusive).

The complement of event S, is an event whose probability equals 1 minus the probability of event S:

\(P(S^C)=1-P(S)\)

\(P(S^C)+P(S)=1\)

If the probability that a TV set you buy is broken is 2%, then the probability that it's not broken equals to:

The probability that the TV set is not broken (\(S^C\)) is the complement of the even that it’s broken (\(S\)) and equals to:

\(P(S^C)=1-P(S)=1-2\%=98\%\)

Note that just as \(S^C\) is the complement of \(S\), \(S\) is the complement of \(S^C\).

The total probability rule can be extended to any number N of mutually exclusive and exhaustive scenarios. The formula for total probability with N scenarios is expressed as follows:

\(P(A)=\sum_{i=1}^nP(AS_i)=\sum_{i=1}^nP(A|S_i)\times P(S_i)\)

- \(P(A)\) - probability of any event \(A\in\Omega\)
- \(S_i\) - mutually exclusive events (scenarios), \(\sum_{i=1}^nS_i=\Omega\)
- \(\Omega\) - sample space (all possible outcomes)
- \(P(A|S_i)\) - conditional probability

Beta&Theta, Inc. manufactures semiconductors. John Delasega, an independent analyst, is trying to estimate the probability that the company’s net profit will increase in the third quarter in comparison with the second quarter. He has assumed that the increase in the net profit of the company is conditional on two scenarios. The probability that the company's net profit increases is 90% if its revenues go up. On the other hand, if the revenues don't increase, the probability that the net profit goes up is 10%. Based on historical data, John has estimated that the probability of an increase in the company’s revenues is 70%.

What's the probability that the company's net profit increases?

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As an analyst, you are trying to estimate a return on stocks of ABC company. You assume three scenarios for the condition of the economy and make the return on the stock conditional on them. You gathered your data in the table:

Probability of a scenario | Probability that return exceeds 10% given a scenario | |
---|---|---|

Scenario 1 (strong growth) | 50% | 90% |

Scenario 2 (modest growth) | 30% | 60% |

Scenario 3 (stagnation) | 20% | 30% |

What is the unconditional probability that the return on a stock of ABC exceeds 10%?

To solve the problem, we need to use the total probability rule. Since there are three scenarios, the total probability rule takes the following form:

\(P(A)=P(A|S_1)\times{P(S_1)}+P(A|S_2)\times{P(S_2)}+P(A|S_3)\times{P(S_3)}=\\=0.9\times0.5+0.6\times0.3+0.3\times0.2=0.45+0.18+0.06=0.69\)

Thus, the probability that the return on a stock of ABC exceeds 10% is 0.69.

- Use the total probability rule to estimate the probability of an event conditional on the occurrence of some possible scenarios.
- The complement of event S, is an event whose probability equals 1 minus the probability of event S.