Probability of not seeing x down days, in y months

This is a simple twist off of the up-streaks we have seen in the financial markets so far this year.  Here below is the start of the simple formulaic logic to solve for this chance occurrence, assuming a random walk without trend.  We start with the probability of three down days.

q = chance of getting a down day is 50%
qqq = chance of getting a down day, three days straight, is 50%^3 = 12.5%

Now each month there are about 22 trading days, or 7 consecutive 3-day periods.  This would be the same probability as solving this problem with rolling 3-day periods since we are, in this case, seeking to solve a streak calculation.  So continuing the formula above, and solving for the chance of not seeing three straight down days in a month implies:

(1-qqq)^7 
= 87.5%^7
= 38%

Then to compound this process of not seeing three straight down days, over a range of months in a row, we see how quickly this probability falls.  See the leaf-like chart below.

Number of straight months | Probability of not seeing 3 down days over this time (50 *'s = 100%)
0 | ************************************************** (100%)
1 | ******************** (38%)
2 | ******** (14%)
3 | *** (6%)
4 | ** (2%)
5 | * (1%)

The chance of not seeing four straight down days is the probability of seeing three straight down days multiplied by half of the fourth days also being down.  So 12.5%/2=6.25%.  Combined with the fact that there are a high 3/4 as many 4-day consecutive periods a month versus 3-consecutive day periods, the probability therefore of not seeing this rare event is even higher!  Or (1-qqqq)^5=70%.  Let's see this compound probability over a range of months, in this leaf-like chart below.

Number of straight months | Probability of not seeing 4 down days over this time (50 *'s = 100%)

0 | ************************************************** (100%)

1 | ************************************ (70%)

2 | ************************* (50%)

3 | ****************** (35%)

4 | ************* (25%)

5 | ********* (17%)

Lastly, the chance of seeing four straight down days is similar to this combinatorics note.  We see that the complement of the "n-1 of n" streaks combinatorics is equivalent to the chance of not "not seeing four straight down days".  We show this convoluted probability over the same range of months as above, in this leaf-like chart below.

Number of straight months | Probability of seeing 4 down days over this time (50 *'s = 100%)

0 | ************************************************** (0%)

1 | ************************************************** (30%)

2 | ************************************************** (50%)

3 | ************************************************** (65%)

4 | ************************************************** (75%)

5 | ************************************************** (83%)