Content in social media stream 
Imagine a social media stream like Instagram. Every day a person is given $8$ brand new posts a day. A post has $33\%$ chance of being a video and $66\%$ chance of being an image. 
Let's say in a given day those $8$ posts can show up at any time during the $24$ hours. 
Now the person is only awake $16$ hours a day and only opens the Instagram app twice a day, but when he/she enters the app they will see $3$ posts. 
  The $3$ posts can be either the new posts or older pre-existing posts. 
On an average day how many new video posts will the person see? On an average day how many new image posts will the person see. Lastly how many old posts will the user see. 

I'm actually not looking for the specific answer but what formula would help me solve this questions if the numbers are moved around. 
Thank you for the help, please feel free to ask any questions.
 A: If the user is going to see the 3 most recent posts, the fact that there are 8 posts a day is irrelevant, and the fact that the user is awake for 16 hours a day is irrelevant.
The probability that the most recent post is video is 1/3
The probability that the 2nd most recent post is video is again 1/3
The probability that the 3rd most recent post is video is again 1/3
Therefore the probability of what they will see:
$Prob(3 videos)  =  (1/3 *1/3 * 1/3)  =  1/27$
$Prob(2 video, 1 images)  =   3 *(1/3 * 1/3 * 2/3)  =  6/27$
$Prob(1 video, 2 images)  =  3 * (1/3 * 2/3 * 2/3)  =  12/27$
$Prob(3 images)  =  (2/3 * 2/3 * 2/3)  =  8/27$
So the average number of videos in the 3 most recent posts:
$3*(1/27) + 2*(6/27) + 1*(12/27) + 0*(8/27) = 1$
Similarly the average number of videos in the 3 most recent posts:
$0*(1/27) + 1*(6/27) + 2*(12/27) + 3*(8/27) = 2$
So on average, a user will see 1 video and 2 images every time he/she opens the app.
So on an average day, the user will see 2 videos and 4 images.
This only becomes complicated if the 2 times the user opens the app are close enough that the 3 most recent posts overlap with the 3 posts he/she saw last time. In the case, you should clarify if you care how many total post types they saw (including duplicates) or unique posts they saw.
