If 25% of the people <em>are</em> vaccinated, then 75% of the people are <em>not</em> vaccinated. Of those not vaccinated, each has a 50% chance of contracting the disease. The probability that someone is both not vaccinated and contracts the disease is (0.75)(0.5)=0.375. The probability that someone is vaccinated and contracts the disease is (0.25)(0.1)=0.025 (it is multiplied by 0.1 because if the vaccine is 90% effective, then there is a 10% chance someone that is vaccinated can contract the disease. Add these together for the total: 0.375+0.025=0.4 There is a 40% chance that someone chosen at random will contract the disease.
Theoretical probability is a method to express the likelihood that something will occur. It is calculated by dividing the number of favorable outcomes by the total possible outcomes.
The empirical probability, relative frequency, or experimental probability of an event is the ratio of the number of outcomes in which a specified event occurs to the total number of trials, not in a theoretical sample space but in an actual experiment.