What Is Probability?

Since inferential statistics deal with information obtained from part of a population to make inferences concerning the entire population, we can never be certain that our inferences are correct; we are dealing with uncertainty.

The science of uncertainty is called probability. Probability is another term for the likelihood of occurrence. We have to make management decisions based on uncertainty on a daily basis. Our need to cope with this uncertainty leads us to study and use the theory of probability.

A random experiment is a process or course of action that generated the uncertain outcomes to which we assign probabilities. Random experiments provide the raw data for statistical analysis.

The first step in assigning probabilities is to determine sample space. The sample space of a random experiment is a list of all the possible outcomes of the experiment.

This list needs to be exhaustive as well as mutually exclusive.

The individual outcomes in a sample space are called simple events.

Example of probability:

An observer stands at the bottom of a freeway and records the turning direction (L or R) of each of 3 successive vehicles.

The sample space contains 8 possible outcomes:

[LLL, RLL, LRL, LLR, RRL, RLR, LRR, RRR]

Each of these outcomes is a simple event. Various combinations are possible:

The event that exactly 1 car turns right = [RLL, LRL, LLR]

The event that no more than 1 car turns right = [LLL, RLL, LRL, LLR]

The event that all cars turn in the same direction = [LLL, RRR]

Approaches To Assigning Probability

The classical approach (theoretical) is used in calculating probabilities when each of the outcomes in the sample space is regarded as being equally likely, for example, in experiments involving tossing fair coins, rolling of the dice or selecting cards from a well-mixed deck. To calculate the probability, we count the number of possible ways the event (E) can occur, also known as a success, and divide by the total number of outcomes in the sample space.

P(E) = No. of successes/Total no. of outcomes

Example:

In drawing a card from a deck of 52 cards, what is the probability that it will be an ace?

P(E) = 4/52 = 0,07

More examples of the classical approach to probability can be found in gambling:

If you roll a 6 sided dice, what is the probability that it will land on a particular number, e.g. 6?

1/6 = 0.16

The relative frequency approach (Experimental) is an alternative way to look at probability by carrying out an experiment to determine the proportion in the long run. If the experiment is repeated n times and the event (E) occurs m times then:

P(E) = m(no. of times E occurs) / n(no. of times experiment repeated)

Example:

An insurance company wants to know what the probability is that an insured owner of a new home, selected at random, will not file a claim in the first year of the policy. A file on the insurance history of 600 new houses is checked and shows that 480 houses did not have an insurance claim filed during the first year of the policy.

P(no claim) = 480/600 = 0.8

In the subjective approach, in some cases, no objective basis exists for determining probabilities, i.e. we have no history of repetitions, for example, the probability that an explosion can occur at a nuclear plant. Asking a series of questions may enable a researcher to find out people’s subjective probabilities, or their degree of confidence in certain events happening.

Probability Rules

We can use various rules of probability to compute the probabilities of more complex, related events.

Click here to view a video that explains Multiplication and Addition Rule, Probability, Mutually Exclusive and Independent Events.

Complement Rule

The sum of the probabilities assigned to the simple events in a sample space must be 1. If the probability of the occurrence of event A is P(A) and the probability that event A may not occur is P(A ̃) then:

P(A) + P(Ã ̃) = 1

P(A) = 1 - P(Ã)

P(Ã ̃) = 1 - P(A)

Despite its simplicity, the complement rule can be very useful. The task of finding the probability that an event will not occur, and then subtracting it from 1, is often easier or less time consuming than the task of directly computing the probability that it will occur.

Example:

If the probability of completing a job is 0.8, the probability of not completing it is

(1 - 0.8) = 0.2.

Special Rule of Addition

This rule states that the probability of event A or event B occurring in a given observation is equal to the probability of event A plus the probability of event B.

P(A or B) = P(A) + P(B)

To apply this rule, the events must be mutually exclusive (they cannot occur at the same time).

Example: If the probability that a manufacturer’s raw material price index will go up during a certain month is 0.82, and the probability that it will remain unchanged is 0.13, then the probability that it will go up or remain unchanged is:

P(increase or unchanged) = P(increase) + P(unchanged)

= 0.82 + 0.13

= 0.95

General Rule of Addition

This rule states that the probability of either event A or event B occurring equals the probability of event A occurring, plus the probability of event B occurring, minus the probability of both occurring.

P(A or B) = P(A) + P(B) - P(A and B)

Apply the rule when outcomes are not mutually exclusive (i.e. if both events may occur simultaneously in a single observation) to avoid double-counting, the probability of both events occurring together is subtracted from the sum of probabilities of each event occurring separately.

Example:

The probability that a person stopping at a garage for petrol will ask to have the tyres checked is 0.12; the probability that a person will ask to have the oil checked is 0.29 and the probability that a person will ask to have both checked is 0.07. The probability that a person stopping at this garage will ask to have.

Either the tyres or the oil checked is:

P(T or O) = P(T) + P(O) - P(T and O)

= 0.12 + 0.29 - 0.07

= 0.34

Neither the tyres nor the oil checked is:

- P(T or O) = 1 - 0.34

= 0.66

Special Rule of Multiplication

This rule states that when A and B are independent, (i.e. they do not influence each other) the probability of both occurring together is the product of individual probabilities.

P(A and B) = P(A) . P(B)

Example:

The probability of any car finishing the Grand Prix is 0.6. In this year 2 cars were entered by the Total Team. The probability that:

Both cars will finish is:

P (F1 and F₂) = P(F^₁ ) × P(F₂)

= 0.6 x 0.6

= 0.36

Neither of the 2 will finish is:

P(F and F) = 0.4 × 0.4

= 0.16

General Rule of Multiplication

This rule is to combine events that are dependent on each other. For 2 events (A and B), the probability of the second event (B) is affected by the outcome of the first event (A). This is often referred to as conditional probability. The probability of both A and B occurring is:

P(A and B) = P(A). P(B/A)

Where P(B/A) is the probability B will occur, given that A has occurred.

Example:

The Ready Tune-up Centre received a shipment of 4 carburettors, and 1 is known to be defective. If 2 are selected at random from the rest and tested, the probability that:

Neither will be defective is:

P(W₁ and W₂ ) = P (W₁ ) . P (W₂/W₁)

= 3/4. 2/3

= 6/12

(Note that if the first one works the probability of the second one working is 2/3.)