**PROBABILITY**

**What is Probability?**

Probability is the measure of the likeliness that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more certain we are that the event will occur.

Probability theory is one of the most widely applicable mathematical theories. It deals with uncertainty and teaches you how to manage it. It is simply one of the most useful theories you will ever learn.

Please do not misunderstand: We are not learning to predict things; rather, we learn to utilise predicted *chances* and make them useful. Therefore, we don’t care about questions like *what is the probability it will rain tomorrow?* But given that the probability is 60% we can make deductions, the easiest of which is *the probability it will not rain tomorrow is 40%*.

As suggested above, a *probability* is a percentage, and it’s between 0% and 100% (inclusive). Mathematicians like to express a *probability* as a *proportion*, i.e. as a number between 0 and 1. So the probability that it will rain tomorrow is 0.6.

**Application**

You might ask why we are even studying probability. Let’s see a very quick example of probability in action.

Consider the following gambling game: Toss a coin; if it’s heads, I give you $1; if it’s tails, you give me $2. You will easily notice that it is not a fair game – the chances are the same (50%-50%) but the rewards are different. Even though we are playing with probability, there are useful, and sometimes not so obvious, conclusions we can make: one of them is that *in the long run* I will become richer and you will become poorer.

Another real-life example: I observed one day that there are dark clouds outside. So I asked myself, should I bring an umbrella? I use my observation of dark clouds as per my usual daily deciding routine. Since in past experiences, dark clouds are early warning signs of rain, I am more likely to bring an umbrella.

In real life, probability theory is heavily used in risk analysis by economists, businesses, insurance companies, governments, etc. An even wider usage is its application as the basis of statistics, which is the main basis of all scientific research. Two branches of physics have their bases tied in probability. One is clearly identified by its name: statistical mechanics. The other is quantum physics.

**Event and Probability**

Roughly, an *event* is something we can assign a *probability* to. For example *the probability it will rain tomorrow is 0.6*; here, the event is *it will rain tomorrow*, and the assigned probability is 0.6. We can write

P(it will rain tomorrow) = 0.6

Mathematicians typically use abstract letters to represent events. In this case we choose *A* to represent the event *it will rain tomorrow*, so the above expression can be written as

P(A) = 0.6

Another example is *a fair die will turn up 1, 2, 3, 4, 5 or 6 with equal probability each time it is tossed*. Let *B* be the event that it turns up 1 in the next toss. We write:

P(B) = 1/6

**Impossible and certain events**

Two types of events are special. One type are the impossible events (e.g., a roll of a die will turn up 7); the other type are certain to happen (e.g., a roll of a die will turn up as one of 1, 2, 3, 4, 5 or 6). The probability of an impossible event is 0, while that of a certain event is 1. We write

P(Impossible event) = 0

P(Certain event) = 1

The above reinforces a very important principle concerning probability. Namely, the range of probability is between 0 and 1. You can **never** have a probability of 2.5! So remember the following

0 ≤ P (E) ≤ 1

for all events E.

**Complement of an event**

A most useful concept is the **complement** of an event. Here we use to represent the *event* that *the die will NOT turn up 1 in the next toss*. Generally, putting a bar over a variable (that represents an event) means the opposite of that event. In the above case of a die:

P(E^{’}) = 5/6

it means *The probability that the die will turn up 2, 3, 4, 5 or 6 in the next toss is 5/6*. Please note that

P(E^{’}) = 1- P(E^{’})

for any event E.