Can someone help make sense of this for me? I understand what an unconditional probability, a conditional probability, and mutually exclusive and collectively exhaustive events are, but I don’t understand how you can have an unconditional probability of an event in terms of conditional probabilities…?? I may be reading to much into this but could someone lend a hand? Thanks!!
This can be sometimes confusing. Let’s start with a definition, followed by an example. A conditional probability is defined as a probability of an event given the occurrence of another event. In contrast, an unconditional probability is a probability of an event occurring without restrictions. Another way to think of it is - conditional has some dependency whereas unconditional is independent. So, if an expectation is such that a conditional event A takes place given an occurrence of another event B, AND the product of these two events, AxB, creates an unconditional result then this is known as the ‘total probability rule’. For example, in hypothetical terms, let’s say google expects to earn $50/share from increased advertising revenue if their total page views exceed 100 trillion times next year. The probability of this happening is .001. And there is a .30 probability that a start-up next year will come up with a better search algorithm than google. The total probability that google will meet its earnings target if the page views are exceeded AND a start-up fails to deliver is 0.0007. Does this help.
It does, that makes much more sense. It is all about how you word those definitions and examples. Thanks a lot for the explanation!
Er… No offense guys. I know everyones only trying to help here and has the noblest of intentions, but Sujan’s explanation and example don’t have anything to do with the total probability rule. The total probability rule expresses unconditional probabilities in terms of conditional probabilities for mutually exclusive and exhaustive events. Suppose that we are concerned with determining the probability of you carrying an umbrella on a given day, P(A). We are given the following information: The probability of rain today = P(S) = 0.4 The probability of no rain today = P(Sc) = 0.6 P(S) plus P(Sc) equals 1 and they are mutually exclusive, exhaustive events. The probability of you carrying an umbrella, P(A), an unconditional probability, equals the sum of two joint probabilities: P(carrying an umbrella and rain) ie. P(AS) ; plus P(carrying an umbrella and no rain) ie. P(ASc) P(carrying an umbrella and rain) can be calculated as P(carrying umbrella given rain) * P(rain) Similarly: P(carrying an umbrella and no rain) can be calculated as P(carrying umbrella given no rain) * P(No rain) Therefore, if you are also provided the following information: The probability of you carrying an umbrella given that it rains = P(A|S) = 0.85 The probability of you carrying an umbrella given that it does not rain = P(A|Sc) = 0.25 You can calculate the probability of carrying an umbrella as: P(A) = P(AS) + P(ASc) P(A) = P(A|S) * P(S) + P(A|Sc) * P(Sc) P(A) = (0.85*0.4) + (0.25*0.6) = 0.49 The total probability rule is a way of calculating an unconditional probability, P(A) using conditional probabilities, P(A|S) and P(A|Sc). The unconditional probabilities, P(A|S) and P(A|Sc), are basically used to calculate the joint probabilities, P(AS) and P(ASc). The sum of P(AS) and P(ASc) give you P(A) if S and Sc are mutually exclusive and exhaustive scenarios. I HATE RESOLVING QUANT ISSUES 
Thank you. That example was very helpful!
Thank you beatthecfa for pointing that out and for the thorough explanation with a textbook example. I agree that I was not entirely clear on the explanations. Although the definition I have used for total probability is incomplete the example used was to show the existence of ‘an unconditional probability of an event in terms of conditional probabilities’ - something aapl was not sure about. The start-up coming up with a better search algorithm is the mutually exclusive and exhaustive scenario. It is pretty much similar to the interest rate examples used in the CFAI books. And I cannot appreciate enough that you took the time to correct my irregularities. Looking forward to more of these in the future!
Sujan… Just one more thing. You said “For example, in hypothetical terms, let’s say google expects to earn $50/share from increased advertising revenue if their total page views exceed 100 trillion times next year. The probability of this happening is .001. And there is a .30 probability that a start-up next year will come up with a better search algorithm than google. The total probability that google will meet its earnings target if the page views are exceeded AND a start-up fails to deliver is 0.0007.” In this example, I do not see any conditional probabilities. Conditional probabilities are ‘given that’ probabilities. The two probabilities that you do mention - probability of page views exceeding 100 trilllion, and the probability that the start up will come up with a better search algorithm, are ‘stand-alone’ probabilities that you are using to calculate a ‘joint probability’. Please understand that my intention here is only to help you all. I am genuinely trying to ensure that your concepts, and those of other visitors to this board are entirely correct. Good luck 
Happy to clarify them further. The decoupling of the two probabilities and reshuffling some verbiage may paint the picture more clearly. Mutually exclusive and exhaustive event >>And there is a .30 probability that a start-up next year will come up with a better search algorithm than google.<< Interpretation: The probability of better search algorithm: P(S) = 0.3 The probability of no better search algorithm: P(Sc) = 0.7 Description: Both P(S) and P(Sc) are mutually exclusive, exhaustive events, and equal 1. Conditional Probability >>google expects to earn $50/share from increased advertising revenue if their total page views exceed 100 trillion times next year. The probability of this happening is .001.<< Interpretation: The probability of earnings of $50/share given page views are exceeded: P(A|S) = 0.001 I realize that ‘S’ in P(A|S) needs to match ‘S’ in P(S) - essentially, the ‘S’ represents the mutually exclusive and exhaustive event, which needs to be exactly the same for both P(A|S) and P(S). Therefore, The probability of earnings of $50/share given start-up with better search algorithm: P(A|S) = 0.001 The probability of earnings of $50/share given start-up with no better search algorithm: P(A|Sc) = 0.2 Description: The probability of earnings of $50/share ‘given that’ the start-up comes up with a better search algorithm, is .001. In contrast, the probability of earnings of $50/share ‘given that’ the start-up does not come up with a better search algorithm, is 0.2. Unconditional Probability >>The total probability that google will meet its earnings target if the page views are exceeded AND a start-up fails to deliver is 0.0007<< Interpretation: The unconditional probablility of earnings: P(A) Description: The probability of meeting earnings target given the page views are exceeded - estimated at .001, times 0.7, the probability that start-up has no better algorithm next year, equals 0.0007. The above description now is redudant given the conditional probability above has changed. The following holds instead: The probability of meeting earnings target given the start-up is successful in a better algorithm - estimated at .001, TIMES 0.3, the probability that start-up has a better algorithm next year, AND The probability of meeting earnings target given the start-up is not successful in a better algorithm - estimated at .2, TIMES 0.7, the probability that start-up has no better algorithm next year, EQUALS the unconditional probability of 0.1403. Calculation: P(A) = P(AS) + P(ASc) P(A) = P(A|S) * P(S) + P(A|Sc) * P(Sc) P(A) = (0.001*0.3) + (.2*0.7) = 0.1403 Sorry about the long post. In my original example, I was missing the 'P(A|Sc) * P(Sc) ’ - this is what I meant by not being entirely clear.