romeo & juliet probability problem the first time that romeo confesses to juliet she responses positively with probability 50%. If she responses negatively he confesses again but with positive response probability 20%. If she responses negatively again he tries one last time with success rate 10%. what is the probability of them hooking up if he is willing to try 3 times?

Romeo gets a yes on first try with probability $0.5$. If he reaches the second stage (probability $0.5$), he gets a yes with probability $0.2$, so that's another $.5 \times .2 = 0.1$ If he reaches the third stage (probability $0.5 \cdot 0.8 = 0.4$), he gets a yes with probability $0.1$, so that's another $.4 \times .1 = 0.04$. Add these up to get $0.5+0.1+0.04 = 0.64$.