PROBABILITY -- PART ONE

Experiment

An experiment is an act for which the outcome is uncertain.

Examples of experiments are rolling a die, tossing a coin, surveying a group of people on their favorite soft drink or beer, etc.

Sample Space

A sample space S for an experiment is the set of all possible outcomes of the experiment such that each outcome corresponds to exactly one element in S. The elements of S are called sample points. If there is a finite number of sample points, that number is denoted n(S), and S is said to be a finite sample space.

For example, if our experiment is rolling a single die, the sample space would be S = {1, 2, 3, 4, 5, 6}.

If our experiment is tossing a single coin, our sample space would be S = {Heads, Tails}.

If our experiment is surveying a group of people on their favorite soft drink, our sample space would be all of the soft drinks or beer on the survey.

Event

Any subset E of a sample space for an experiment is called an event for that experiment.

For example, if our experiment is rolling a single die, an event E could be rolling an even number, thus E = {2, 4, 6}.

If our experiment is tossing a single coin, an event E could be tossing a Tail, where E = {Tails}.

If our experiment is surveying a group of people on their favorite soft drink, an event E could be picking a diet soft drink.

If beer, an event E could be picking one brand of beer.

Giacomo wrote:Empirical Probability

Finding the probability of an empirical event is specifically based on direct observations or experiences.

For example, a survey may have been taken by a group of people. If the data collected is used to find the probability of an event tied to the survey, it would be an empirical probability. Or if a scientist did research on a topic and recorded the outcome and the data from this is used to find the probability of an event tied to the research, it would also be an empirical probability.

Empirical Probability Formula :

P(E) = (number of times event E occurs) / (total number of observed occurences)

P(E) represents the probability that an event, E, will occur.

The numerator of this probability is the number of times or ways that specific event occurs.

The denominator of this probability is the overall number of ways that the experiment itself could occur.

Equiprobable Space

A sample space S is called an equiprobable space if and only if all the simple events are equally likely to occur.

Some quick examples of this are:

A toss of a fair coin. It is equally likely for a head to show up as it is for a tail.

Select a name at random from a hat. Since it is at random, each name is equally likely to be picked.

Throwing a well balanced die. Each number on the die has the same amount of chance of coming up.

Theoretical Probability

Theoretical probability is finding the probability of events that come from an equiprobable sample space or, in other words, a sample space of known equally likely outcomes.

For example, finding various probabilities dealing with the roll of a die, a toss of a coin, or a picking of a name from a hat.

Theoretical Probability Formula :

P(E) = (number of outcomes in E) / (number of outcomes in S) = n(E)/n(S)

If E is an event of sample space S, where n(E) is the number of equally likely outcomes of event E and n(S) is the number of equally outcomes of sample space S, then the probability of event E occurring can be found using the Theoretical Probability Formula above.

Mutually Exclusive

In general, events E and F are said to be mutually exclusive if and only if they have no elements in common.

For example, if the sample space is rolling a die, where S = {1, 2, 3, 4, 5, 6},

and E is the event of rolling an even number, E = {2, 4, 6}

and F is the event of rolling an odd number, F = {1, 3, 5},

E and F are mutually exclusive, because they have NO elements in common.

Now let's say that event G is rolling a number less than 4, G = {1, 2, 3}.

Question : Would event G and E be mutually exclusive?

If you say no, you are correct, they have one element, the number 2, in common. G and F would not be mutually exclusive either.

Properties of Probability

Property #1: 0 =< P(E) =< 1 (=< means equal or less)

The probability can be equal to 0 or less or equal to 1, it will never exceed 1, since the bottom number of the probability is the total number - which is the highest number.

Property #2: P(not E) = 1 - P(E)

N.B. I will use the symbol "~" for not. So, ~ E means not E. We can now write:

P(~ E) = 1 - P(E)

Property #3: P(E) = 1 - P(~ E)

This is just like Property 2 in reverse.

Property #4: "Or" probabilities with mutually exclusive events

P(E OR F) = P(E) + P(F)

We could use the symbol "\/" for OR. We could rewrite it as follows:

P(E \/ F) = P(E) + P(F)

Since we are dealing with sets that are mutually exclusive, this means they have no elements in common. So we can just add the two probabilities together without running a risk of having something counted twice.

Property #5: "Or" probabilities with events that are NOT mutually exclusive

P(E OR F) = P(E) + P(F) - P(E AND F)

We could use the symbol "/" for AND. We could then rewrite it as follows:

P(E \/ F) = P(E) + P(F) - P(E /\ F)

Since we are dealing with events that are NOT mutually exclusive, we run the risk of elements being counted twice if we just add them together as in Property 4 above.

You need to subtract the intersection to get rid of the elements that were counted twice. In other words, you may have some elements in common, so if we add the number of elements in E to the number of elements of F, we may be adding some elements twice, so to avoid this we need to subtract the number of elements in the intersection of the two events - which would be all the elements that are in both sets.

Independent Events

Two events are independent of each other if the outcome of one event does not affect the outcome of the other event.

E and F are Independent Events if an only if

P(E AND F) = P(E)*P(F)

P(E /\ F) = P(E)*P(F)