Law of Total Probabilities

Exhibit 33.5 The events H (heart), D, C and S are mutually exclusive and collectively exhaustive.

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, A1, A2 … An, the probability for any event B to occur:

$$P(B)=\sum_{i=1}^n P(B∩A_i )=P(B∩A_1 )+P(B∩A_2 )… P(B∩A_n)$$ $$P(B)=P(B│A_1 )P(A_1 )+P(B│A_2 )P(A_2 )… P(B|A_n)P(A_n)$$

Taking cards for example (Exhibit 33.5), 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 33.6, 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 33.6.

Exhibit 33.6 Computing the weighted average brand buyers.

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