Probability theory provides the foundation for quantitative marketing analytics and research. This introduction provides an overview that touches on some of the key terms and concepts.

A *random variable* is a variable whose possible values are numerical
outcomes of a probabilistic experiment. It can either be discrete, taking a finite
set of numeric values, or continuous, in which case the range of values is infinite.

*Probability* is the likelihood that an event will occur. It varies
between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The higher
the probability, the more likely that the event will occur.

Probability is defined in the context of a sample space or *universe*, i.e.,
a set of all possible outcomes. For instance, the universe of stores in retail tracking,
is all the stores that represent the “total market” measured by the tracking service.

As an example, consider pack of cards. The universe constitutes
the 52 cards shown in Exhibit 33.1. If you pick a card at random, the probability it
will be a queen, P(Q), is 4/52, and the probability it will be the queen of hearts,
P(Q_{hearts}) is 1/52.

The *complement* of an event, Q, is the probability that the event will
not occur. It is denoted by: Q̄: P(Q̄)=1−P(Q).

The probability that the card you pick is not a queen, is 1 – P(Q) = 48/52.

As shown in Exhibit 33.2, the events picking a queen (Q) and picking a king (K)
are *mutually exclusive*; their range of possible values do not overlap. For such *mutually
exclusive* events, the probability that either one or the other will occur, is a summation of their
individual probabilities:

The probability that the card is either a queen or a king is 4/52 + 4/52 = 8/52.

If Q and H are not mutually exclusive events, their range of possible values will overlap as shown in Exhibit 33.3, and the probability that either will occur is:

$$P(Q \,or \,H) = P(Q) + P(H) - P (Q \,and \,H)$$The probability that the card is either a queen or a heart is 4/52 + 13/52 – 1/52.

Generally speaking, there are two types of probabilities, objective and subjective:

*Objective probabilities*that can be estimated from proportions, based on data collected. For instance, P(Q), or for instance, the probability that a retail store is handling the brand Coca-Cola.*Subjective probabilities*that*cannot*be estimated from long-run proportions. For example, an applicant feels she has a 50% chance of getting a job. This is her opinion, and it is subjective.

Conditional Probability is the probability of an event given that another event has already occurred. For instance, probability that the card is a queen, given it is a heart.

$$P(Q|H) = \text{probability that Q will occur, given H has occurred.}$$ $$P(Q│H)=(P(Q∩H))/(P(H))=(1/52)/(13/52)$$This implies:

$$P(Q∩H)=P(Q│H)×P(H)=P(H│Q)×P(Q)$$Events are independent if knowing whether one has occurred does not change the probability of the other.

Suppose you pick a card and toss a coin. The probability of getting a tail, P(T), in no way influences the probability of getting a queen, P(Q), from a pack of cards.

$$P(T│Q)=P(T),\,and \,P(Q│T)=P(Q)$$ $$P(T∩Q)=P(T)×P(Q)$$The law of total probability is a rule relating marginal probabilities to conditional probabilities. It expresses the total probability of an outcome which can be realized via several distinct events.

Consider *n* mutually exclusive and collectively exhaustive events, *A _{1},
A_{2} … A_{n}*, the probability for any event

Taking cards for example (Exhibit 33.4), the probability of picking a queen:

$$P(Q)=P(Q│H)P(H)+P(Q│D)P(D)+P(Q|C)P(C)+P(Q|S)P(S)$$Basically, this is the same as weighted average.

*Example:* As shown is Exhibit 33.5, the age profile of a target population
(universe) is broken down into 4 groups: below 20, 20 to 30, 30 to 50 and over 50. The proportion of the
population that falls in each group is 20%, 20%, 40% and 20% respectively, and the proportion of brand
buyers in each is 10%, 15%, 20% and 15%. Based on this information, the proportion of brand buyers in the
target population is 16%, as computed in Exhibit 33.5.

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