A company is deciding on installing a 3-component or 5-component satellite system. It is known that the system functions on any given day if a majority of the satellites components function on that day. On a rainy day, each of the components independently functions with a probability of 0.4, whereas on a dry day, they each independently function with probability of 0.8.
In that particular region, if it is twice as likely to have a rainy day as it is to have a dry day, should the company install the 3-component or 5-component system?
Intuition would tell me to pick the 3 component system because it is twice as likely to rain as shine, and with the probability of a component working so low when it rains, 2/3 given rain should be easier than 3/5 given rain but I'm not sure how to do this mathematically.