Mathematics of Finance and Elementary Probability

Random Experiment

Let’s say that you toss a fair coin (not a prank coin ). There are only two possible outcomes – a head or a tail. Also, it is impossible to accurately predict the outcome (a head or a tail). In mathematical theory, we consider only those experiments or observations, for which we know the set of possible outcomes. Also, it is important that predicting a particular outcome is impossible. Such an experiment, where we know the set of all possible results but find it impossible to predict one at any particular execution, is a random experiment.

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Even if a random experiment is repeated under identical conditions, the outcomes or results may fluctuate or vary randomly. Let’s look at another example – you take a fair dice and roll it using a box.

When the dice lands there are only six possible outcomes – 1, 2, 3, 4, 5, or 6. However, predicting which one will occur at any roll of the dice is completely unpredictable.

random experiment

                                                                                                                                                 Source: Maxpixel

Further, in any experiment, there are certain terms that you need to know:

  • Trial – A trial is the performance of an experiment.
  • Outcomes – Whenever you perform an experiment, you get an outcome. For example, when you flip a coin, the outcome is either heads or tails. Similarly, when you roll a dice, the outcome is 1, 2, 3, 4, 5, or 6.
  • Event – An event is a collection of basic outcomes with specific properties. For example, ‘E’ is the event where our roll of a six-sided dice has an outcome of less than or equal to 3. Therefore, E is the collection of basic outcomes where the result is 3 or less. Symbolically, E = {O1, O2, O3}. It is important to note that depending on the event, the outcomes can be of any number (even zero).

Random Experiment – Types of Events

In a random experiment, the following types of events are possible:

random experiment

Simple and Compound Events

Simple or Elementary events are those which we cannot decompose further. For example, when you toss a coin, there are only two possible outcomes (heads or tails).

The event that the toss turns up a ‘head’ is a simple event and so is the event of it turning up a ‘tail’. Similarly, when you roll a six-sided dice, then the event that number 3 comes up is a simple event.

The Compound or Composite events are those which we can decompose into elementary or simple events. In simpler words, an elementary event corresponds to a single possible outcome of an experiment.

On the other hand, a compound event is an aggregate of some elementary events and we can decompose it into simple events.

To give you some examples, when you toss a fair coin, the event ‘turning up of a head or a tail’ is a compound event. This is because we can decompose this event into two simple events – (i) turning up of the head and (ii) turning up of the tail.

Similarly, when you roll a six-sided dice, the event that an odd number comes up is a compound event. This is because we can break it down into three simple events – (i) Number 1 comes up, (ii) Number 3 comes up, and (iii) The third odd number 5 comes up.

Learn more about Random Variables here in detail.

Equally Likely Events

If among all possible events, you cannot expect either one to occur in preference in the same experiment, after taking all conditions into account, then the events are Equally Likely Events.

Examples of Equally Likely Events

Back to our favorite coin. Tossing a fair coin has two simple events associated with it. The coin will turn up a ‘head’ or a ‘tail’. Now, there is an equal chance of either turning up and you cannot expect one to turn up more frequently than the other. Also, in the case of rolling a six-sided dice, there are six equally likely events.

Mutually Exclusive Events

In a random experiment, if the occurrence of one event prevents the occurrence of any other event at the same time, then these events are Mutually Exclusive Events.

Examples of Mutually Exclusive Events

Let’s call the coin back into action, shall we?

When you toss a fair coin, the turning up of heads and turning up of tails are two mutually exclusive events. This is because if one turns up, then the other cannot turn up in the same experiment.

Similarly, when you roll a six-sided dice, there are six mutually exclusive events.

Remember, mutually exclusive events cannot occur simultaneously in the same experiment. Also, they may or may not be equally likely.

Independent Events

Two or more events are Independent Events if the outcome of one does not affect the outcome of the other. For example, (coin again!) if you toss a coin twice, then the result of the second throw is not affected by the result of the first throw.

Dependent Events

Two or more events are Dependent Events if the occurrence or non-occurrence of one in any trial affects the probability of the other events in other trials.

Examples of Dependent Events

No coin this time.

Let’s say that the event is drawing a Queen from a pack of 52 cards. When you start with a new deck of cards, the probability of drawing a Queen is 4/52. However, if you manage to draw a Queen in one trial and do not replace the card in the pack, then the probability of drawing a Queen in the remaining trials becomes 3/51.

Exhaustive Events

In order to understand Exhaustive Events, let’s take a quick look at the concept of Sample Space.

Sample Space

The Sample Space (S) of an experiment is the set of all possible outcomes of the experiment.

Going back to our coin, the sample space is S = {H, T} … where H-heads and T-tails. Similarly, when you roll a six-sided dice, the sample space is S = {1, 2, 3, 4, 5, 6}. Every possibility is a sample point or element of the sample space.

Further, an event is a subset of the sample space and can contain one or more sample points. For example, when you roll a dice, the event that an odd number appears has three sample points.

Coming back to exhaustive events, the total number of possible outcomes of a random experiment form an exhaustive set of events. In other words, events are exhaustive if we consider all possible outcomes.

Solved Question

Q1. What is a Random Experiment?

Answer: An experiment, where we know the set of all possible results but find it impossible to predict one at any particular execution, is a random experiment.

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