I am having problems answering this exercise, what is the thinking process that i have to adopt? how do i apply the binomial distribution?
Problem: It is known from the past experience that the probability that a startup company is successful is 0.2 if its business is hard science based and 0.1 if it is not. It is also known that 1/4 of the startups are hard science based. Determine the probability that a startup is successful.
In this case you will need to use weighted probabilities. The answer to your problem is $$P(successful)=0.25P(successful|science-based)+0.75P(successful|non-science-based)$$ You already stated in the given of the problem that $$P(successful|science-based)=0.2$$ and $$P(successful|non-science-based)=0.1$$ So substitute in to get the final answer: $$P(successful)=0.25(0.2)+0.75(0.1)$$ $$P(successful)=0.05+0.075$$ $$P(successful)=0.125=\frac{1}{8}$$ Which is the final answer.