Probability that it will rain at least 6 days = probability that it will rain 6 days + probability it will rain all 7 days
Probability it will rain 6 days .....we want to choose 6 of any 7 days.......and, on each of those days, the probability of rain is 90%.....and on the remaining day, the probability is only 10%.......so......we have this binomial probability:
C(7,6) (.90)^6(.10) = about 37.2%
Probability it will rain all 7 days = (.9)^7 = about 47.8%
So.....summing the probabilities, we have [ 37.2 + 47.8]% = about 85%