R: Simulating coin flips

Here is a problem I thought of:

• Suppose I am watching someone flip a fair coin. Each flip is completely independent from the previous flip.
• I watch this person flip 3 consecutive heads.
• I interrupt this person and ask the following question: If the next flip results in a "head", I will buy you a slice of pizza. If the next flip results in a "tail", you will buy me a slice of pizza. Who has the better odds of winning?

I wrote the following simulation using the R programming language. In this simulation, a "coin" is flipped many times ("1" = HEAD, "0" = TAILS). We then count the percentage of times HEAD-HEAD-HEAD-HEAD appears compared to HEAD-HEAD-HEAD-TAILS:

#load library
library(stringr)

#define number of flips
n <- 10000000 

#flip the coin many times
flips = sample(c(0,1), replace=TRUE, size=n)

#count the percent of times HEAD-HEAD-HEAD-HEAD appears 
str_count(paste(flips, collapse=""), '1111') / n

0.0333663

#count the percent tof times HEAD-HEAD-HEAD-TAIL appears
str_count(paste(flips, collapse=""), '1110') / n

0.062555

From the above analysis, it appears as if the person's luck runs out: after 3 HEADS, there is a 3.33% chance that the next flip will be a HEAD compared to a 6.25% chance the next flip will not be a HEAD (i.e. TAILS).

Thus, could we conclude: Even though the probability of each flip is independent from the previous flip, it becomes statistically more advantageous to observe a sequence of HEADS and then bet the next flip will be a TAILS. Thus, the longer the sequence of HEADS you observe, the stronger the probability becomes of the sequence "breaking".

My Question: Is the R code I have written correct? Does it actually correspond to this problem I have created?

Thanks