Probability of an exact number events over several periods

Question

The probability of an event occuring N times in a day is $ P[N=n] = \frac{1}{2^(n)} where 0\le n $. The number of times an event ocurrs in one day is independent of the number of times the event has happened on any past day.

What is the probability that the event will occur exactly 5 times during any two-day period?

Answer 1

There are six combinations of events that add to 5 over the two day period. These six events occur with the probability below.

$ 2[(1)(\frac{1}{2^5}) + (\frac{1}{2})(\frac{1}{2^4}) + (\frac{1}{2^2})(\frac{1}{2^3})]$

$= \frac{3}{2^4}$

$=\frac{3}{16}$