A vaccine has 90% probability of being effective in preventing a certain disease. the probability of getting the disease if the person is not vaccinated is 50%. in certain geographic region, 25% of the people get vaccinated. if a person is selected at random , find the probability that he or she will contract the disease

If 25% of the people are vaccinated, then 75% of the people are not 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.