Hey guys, I've been working on this problem for a little while and thought i'd try for some external assistance:
So far i've got:"On any given night, there is a 75% probability that you will watch a movie. Of the movies you watch, 60% are Good and 40% are Bad. Your decision to watch a movie or not, as well your choice of title, varies independently from night to night.
1. Given that you watched a total of eight movies in the last ten nights, what is the conditional probability that you watched exactly ﬁve good movies in the ﬁrst seven nights?
2. What is the probability that, starting tomorrow night, the ﬁrst string of movies that you watch (i.e., followed by an no-movie night) will consist of good movies only?"
p[g]=p[good movie] = p[m]*p[g|m] = .75*.6 = .45
p[b] = .75 * .4 = .3
Then I thought to calculate the individual:
p[x] = p[8 movies in 10 nights] = .75^8 * .25^2 * 10!/(8!2!) = .2816
p[y] = p[5 g's in 1st 7] = .45^5*.55^2*7!/(5!2!) = .118
I'm not quite sure where to go from there though for 1. Should i be trying to calculate p[x ∩ y] for expressing it as P = p[x ∩ y] / p[y] ?
For 2. I'm torn between two solutions. The first is just saying that you can focus on the first night that it is NOT a no movie or good movie event, in which case it would be .25/(.3 + .25) = .4545, or do you also have to multiply this by the probability that the first movie night is a good movie, giving .6 * .4545 = .2727?
Any help would be greatly appreciated!