Flipping Coins : Probability of Sequences vs Probability of Individuals 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 make the following offer: 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.

My Question: 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 <- 10^7

#flip the coin many times
set.seed(1)
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

0.0624983

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"?

Ah my friend, you are making a very simple mistake.  In your simulation, you are computing the proportion of times a person could flip 4 heads in a row.  But that is not what you have wagered.
You enter the bet having seen the three heads and have wagered only on the result of the next flip.  Because each flip is independent, and the coin assumed fair, the probability of a heads is the same as a tails and hence the odds are even!
It would have been different had you made the wager at the beginning of the four flips.  In such a case, we could just compute the binomial density.  We would see that 4 heads in a row (conditioned on making only four flips) is very small and so you would have the better odds, again assuming the coin is fair.  But having already seen the 3 flips and then wagering is akin to just betting on a coin flip.
