2.2 Bayes Theorem Applied
Let us return the example of the investor. From theory of binomial distributions, if the probability of some event occurring on any one trial is p, then the probability of x such events occurring out of n trials is expressed as:
For example, the likelihood that 5 out of 20 people will support her enterprise should her location actually fall into the category where 20% of franchises actually achieve 25% saturation is:
The likelihood of the other situations can also be determined:
Event (Market Saturation) p_{i} |
Prior Probability P_{0}(p_{i}) |
Likelihood of Situation P(x=5|p_{i}) |
Joint Probability of Situation |
Posterior Probability |
---|---|---|---|---|
0.10 | 0.05 | 0.03192 | 0.001596 | 0.00959 |
0.15 | 0.05 | 0.10285 | 0.005142 | 0.00309 |
0.20 | 0.20 | 0.17456 | 0.034912 | 0.20983 |
0.25 | 0.20 | 0.20233 | 0.040466 | 0.24321 |
0.30 | 0.40 | 0.17886 | 0.071544 | 0.43000 |
0.35 | 0.10 | 0.12720 | 0.012720 | 0.07645 |
Totals | 1.00 | 0.81772 | 0.166381= P(x=5) |
0.99997 |
The sum of all the Joint Probabilities provides the scaling factor found in the denominator of Bayes Theorem and is ultimately related to the size of the sample. Had the sample been greater than 20, the relative weighting between prior knowledge and current evidence would be weighted more heavily in favour of the latter. The Posterior Probability column of Table 4 shows the results of the Bayesian theorem for this case. By adding up the relative posterior probabilities for market shares >=25% and those <25%, our investor will see that there is a 75% probability that her franchise will make money--definitely a more attractive situation on which to base an investment decision.