###### Bayes' Formula

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Probability is a measure of the likelihood that something will happen. We usually express probabilities as percentages, from 0% (impossible to happen) to 100% (guaranteed).

We can express almost any event as a probability. For instance, we can gauge the likelihood that it will rain on a given day or the likelihood of one passing one’s examination.

Estimation of the probabilities of anticipated outcomes is critical in financial management. More precisely, concepts in probability help analysts quantify risks. The following are the major probability concepts:

A random variable is any quantity whose expected future value is not known in advance. Examples include a mutual fund’s expected future value, the expected dividend payment on a stock, or the expected standard deviation of investment returns. We cannot know the exact future value of an investment since it is usually a function of multiple factors, some of which may be beyond the control of an analyst.

Note that returns that are fixed cannot be described as random variables. For instance, if a risk-free government bond is quoted at a fixed discount of, say, 6%, the bond’s future value can be calculated in advance. As such, it is not a random variable.

An outcome is any possible value that a random variable can take. So, for example, if you roll a six-sided die, there are six possible outcomes since any number, from 1 to 6, can come up.

If a ‘3’ comes up, then that is an outcome. In case a ‘5’ comes up, it is another outcome. So, for example, if a stock offers shareholders a $2 dividend per share at the end of a year, then the $2 dividend is an outcome.

A single outcome or a set of outcomes is known as an event. If we take the die example above, rolling a 4 is an event, rolling an odd number is another event, etc.

Mutually exclusive events are such that one event precludes the occurrence of all the other events. For example, if you roll a die and a 4 comes up, that particular event precludes all the other events, i.e., { 1, 2, 3, 5, and 6 }. In other words, rolling a 1 and a 5 are mutually exclusive events: they cannot occur simultaneously.

Furthermore, there is no way a single investment can have more than one arithmetic mean return. As such, an arithmetic return of, say, 20% constitutes a mutually exclusive event.

Events are said to be exhaustive if they include all possible outcomes. Suppose we roll a die once and categorize the outcomes into two events as follows:

{1, 3, 5} {2, 4, 6}

Each of the above sets of outcomes is an event. The two events are exhaustive because they include all the possible outcomes.

We do a random experiment of drawing one card from a deck. It consists of a sample space of 52 elements (i.e., cards).

A pack or deck of playing cards has 52 cards which are divided into four categories as Spades (♠), Clubs (♣), Hearts (♥), and Diamonds (♦). Each of these categories has 13 cards, 9 cards numbered from 2 to 10, an Ace, a King, a Queen, and a Jack. Hearts and Diamonds are red-faced cards, whereas Spades and Clubs are black-faced cards. Kings, Queens, and Jacks are called face cards.

- A
**random variable**, in this case, will result from drawing a card from a deck of shuffled cards since there is uncertainty in regard to the card that might be drawn. - Suppose a card is drawn randomly from a deck and turns out to be an Ace; then, in this case,
**outcome**is “Ace.” - Drawing two spades and five hearts from the deck qualifies to be an
**event**.

QuestionWhich of the following is themost appropriateterm used for events that cover all the possible outcomes?

- Exhaustive events.
- Independent events.
- Mutually exclusive events.
The correct answer is

A.Exhaustive events are events that cover all possible outcomes. In probability theory and logic, a set of events is jointly or collectively exhaustive if at least one of the events must occur. For example, when rolling a six-sided die, the outcomes 1, 2, 3, 4, 5, and 6 are collectively exhaustive, because they encompass the entire range of possible outcomes.

B is incorrect. Independent events are events that are not affected by the outcome of previous events. For instance, when tossing a coin, the probability of getting a head or tail does not, in any way, depend on whether you got a head or tail on the first toss.

C is incorrect. Mutually exclusive events are events both of which cannot occur simultaneously. For example, when tossing a coin, you can either get a head or tail; there is no possibility of getting both head and tail simultaneously.