Examples

Frequentist example

P(\text{Rare}|\text{Pattern}) &= \frac{P(\text{Pattern}|\text{Rare})P(\text{Rare})} {P(\text{Pattern}|\text{Rare})P(\text{Rare}) \, + \, P(\text{Pattern}|\text{Common})P(\text{Common})} \\ &= \frac{0.98 \times 0.001} {0.98 \times 0.001 + 0.05 \times 0.999} \\ &\approx 1.9\%.

Coin flip example

P(\text{Biased coin}) &= \frac{1}{3} \\ P(\text{Fair coin}) &= \frac{2}{3} \\ P(\text{H}|\text{Fair coin}) &= \frac{1}{2} \\ P(\text{HHH}|\text{Fair coin}) &= \frac{1}{8} \\ P(\text{HHH}|\text{Biased coin}) &= 1 \\ P(\text{Biased coin}|\text{HHH}) &= \frac{P(\text{HHH}|\text{Biased coin})P(\text{Biased coin})}{P(\text{HHH}|\text{Biased coin})P(\text{Biased coin}) + P(\text{HHH}|\text{Fair coin})P(\text{Fair coin})} \\ &= \frac{1 \times \frac{1}{3}}{1 \times \frac{1}{3} + \frac{1}{8} \times \frac{2}{3}} \quad = \quad \frac{\frac{1}{3}}{\frac{10}{24}} \quad = \quad \frac{4}{5}

Drug testing

P(\text{User}|\text{+}) &= \frac{P(\text{+}|\text{User}) P(\text{User})}{P(\text{+}|\text{User}) P(\text{User}) + P(\text{+}|\text{Non-user}) P(\text{Non-user})} \\ &= \frac{0.99 \times 0.005}{0.99 \times 0.005 + 0.01 \times 0.995} \\ &\approx 33.2\%