In the previous posts we have touched upon the concepts of time value of money, NPV and IRR decison rules and basic statistical concepts. From this post we start a series of posts covering probability math and probability distribution. So let’s get started.
Probability concepts relates to the chance of an event occurring. An event is an outcome. If I flip a coin getting a head is an outcome/event. Getting a head and getting a tail are mutually exclusive and exhaustive events i.e. they both cannot occur at the same time and both together are a set of all possible outcomes of flipping a coin.
The probability of occurrence of any event lies between 0 and 1 i.e., between 0% and 100%. The probabilities of occurrence of mutually exclusive and exhaustive events sum to 1. Eg: If I call heads the probability of heads occurring is 50%. Similarly the probability of tails occurring is also 50%. If I call heads 7 times and heads occurs all 7 times, when I call heads the eighth time the probability of heads occurring the 8th time is still 50%. The probability of occurrence of all possibilities of flipping a coin which are mutually exclusive and exhaustive events is therefore 100%.
Stating odds that an event will occur or not occur is another way of stating probabilities in quantitative analysis. Suppose the probability of occurrence of an event is 0.125, i.e. 1/8thof the times an event will occur. The other way of saying that is there is 1 chance out of 8 chances that the event will occur and 7 chances that the event will not occur. Therefore the odds of the event occurring is 1:7. The odds that the event will not occur are therefore the reverse of this i.e. 7:1.
Conditional probability and Unconditional probability:
Conditional probability is when the occurrence of an event is conditional on the occurrence of another event. Eg: If the interest rates are increased the stock market will fall. Therefore the probability of stock market falling is conditional given that interest rate hike has occurred.
Unconditional probability is when the occurrence is of an event is regardless of the occurrence of any other event.
Multiplication and Addition rules:
The joint probability of two events is given by the multiplication rule as,
P(AB) = P(A/B) * P(B)
It reads as
P(AB) = Probability of A and B occurring
P(A/B)= conditional probability that A occurs given that B has occurred.
P(B)=unconditional probability that B has occurred
The whole sentence will the therefore read as Probability of A and B occurring is equal to the probability of A occurring given that B has occurred and B has occurred. The multiplication sign in the equation stands for ‘AND’.
The probability of at least one of two events occurring is given by the addition rule as;
P(A or B) = P(A) + P(B) – P(AB)
The plus sign in the equation stands for ‘OR’ and the equation reads as;
P(A or B) = Probability of A occurring or B occurring
P (A) + P(B) = Probability of A occurring or B occurring
P(AB) = Joint probability of A and B occurring
The whole sentence reads as Probability of A or B occurring is equal to A occurs or B occurs minus the probability that both occur. The probability of both occurring together is deducted to remove the effect of double counting. For mutually exclusive events where joint probability is zero, P(A or B) = P(A) + P(B)
Independent and Dependent events:
Independent events are those where the occurrence of one event has no influence on the occurrence of the other events. An event is independent if;
P(A/B) = P(A)
That is to say, only if the conditional probability of A occurring is equal to the probability of A occurring for certain, A can be called an independent event.
If events are not independent of each other they are called dependent events.
that’s all in this post folks…in the next post we will cover covariance and correlation and permutations and conbinations…so stay posted…bye
For solved examples please refer to the CFA Institute Books and Schweser CFA notes . The problems can be easily solved using the CFA institute approved financial calculators. Please refer to the CFA exam policy and CFA calculator guide.